Combining Philosophers

All the ideas for PG, Francis Bacon and Stewart Shapiro

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373 ideas

1. Philosophy / B. History of Ideas / 3. Greek-English Lexicon

11300	Agathon: good [PG]
	Full Idea: Agathon: good, the highest good
	From: PG (Db (lexicon) [c.1001 BCE], 01)

11301	Aisthesis: perception, sensation, consciousness [PG]
	Full Idea: Aisthesis: perception, sensation, consciousness
	From: PG (Db (lexicon) [c.1001 BCE], 02)

11302	Aitia / aition: cause, explanation [PG]
	Full Idea: Aitia / aition: cause, explanation
	From: PG (Db (lexicon) [c.1001 BCE], 03)
	A reaction: The consensus is that 'explanation' is the better translation, and hence that the famous Four Causes (in 'Physics') must really be understood as the Four Modes of Explanation. They then make far more sense.

11303	Akrasia: lack of control, weakness of will [PG]
	Full Idea: Akrasia: lack of control, weakness of will
	From: PG (Db (lexicon) [c.1001 BCE], 04)
	A reaction: The whole Greek debate (and modern debate, I would say) makes much more sense if we stick to 'lack of control' as the translation, and forget about weakness of will - and certainly give up 'incontinence' as a translation.

11304	Aletheia: truth [PG]
	Full Idea: Aletheia: truth
	From: PG (Db (lexicon) [c.1001 BCE], 05)

11305	Anamnesis: recollection, remembrance [PG]
	Full Idea: Anamnesis: recollection, remembrance
	From: PG (Db (lexicon) [c.1001 BCE], 06)
	A reaction: This is used for Plato's doctrine that we recollect past lives.

11306	Ananke: necessity [PG]
	Full Idea: Ananke: necessity
	From: PG (Db (lexicon) [c.1001 BCE], 07)

11307	Antikeimenon: object [PG]
	Full Idea: Antikeimenon: object
	From: PG (Db (lexicon) [c.1001 BCE], 08)

11375	Apatheia: unemotional [PG]
	Full Idea: Apatheia: lack of involvement, unemotional
	From: PG (Db (lexicon) [c.1001 BCE], 09)

11308	Apeiron: the unlimited, indefinite [PG]
	Full Idea: Apeiron: the unlimited, indefinite
	From: PG (Db (lexicon) [c.1001 BCE], 10)
	A reaction: Key term in the philosophy of Anaximander, the one unknowable underlying element.

11376	Aphairesis: taking away, abstraction [PG]
	Full Idea: Aphairesis: taking away, abstraction
	From: PG (Db (lexicon) [c.1001 BCE], 11)

11309	Apodeixis: demonstration [PG]
	Full Idea: Apodeixis: demonstration, proof
	From: PG (Db (lexicon) [c.1001 BCE], 12)

11310	Aporia: puzzle, question, anomaly [PG]
	Full Idea: Aporia: puzzle, question, anomaly
	From: PG (Db (lexicon) [c.1001 BCE], 13)

11311	Arche: first principle, the basic [PG]
	Full Idea: Arché: first principle, the basic
	From: PG (Db (lexicon) [c.1001 BCE], 14)
	A reaction: Interchangeable with 'aitia' by Aristotle. The first principle and the cause are almost identical.

11312	Arete: virtue, excellence [PG]
	Full Idea: Areté: virtue, excellence
	From: PG (Db (lexicon) [c.1001 BCE], 15)
	A reaction: The word hovers between moral excellence and being good at what you do. Annas defends the older translation as 'virtue', rather than the modern 'excellence'.

11313	Chronismos: separation [PG]
	Full Idea: Chronismos: separation
	From: PG (Db (lexicon) [c.1001 BCE], 16)

11314	Diairesis: division [PG]
	Full Idea: Diairesis: division, distinction
	From: PG (Db (lexicon) [c.1001 BCE], 17)

11315	Dialectic: dialectic, discussion [PG]
	Full Idea: Dialectic: dialectic, discussion
	From: PG (Db (lexicon) [c.1001 BCE], 18)

11316	Dianoia: intellection [cf. Noesis] [PG]
	Full Idea: Dianoia: intellection, understanding [cf. Noesis]
	From: PG (Db (lexicon) [c.1001 BCE], 21)

11317	Diaphora: difference [PG]
	Full Idea: Diaphora: difference
	From: PG (Db (lexicon) [c.1001 BCE], 22)

11318	Dikaiosune: moral goodness, justice [PG]
	Full Idea: Dikaiosune: moral goodness, justice
	From: PG (Db (lexicon) [c.1001 BCE], 23)
	A reaction: Usually translated as 'justice' in 'Republic', but it is a general term of moral approbation, not like the modern political and legal notion of 'justice'. 'Justice' actually seems to be bad translation.

11319	Doxa: opinion, belief [PG]
	Full Idea: Doxa: opinion, belief, judgement
	From: PG (Db (lexicon) [c.1001 BCE], 24)

11320	Dunamis: faculty, potentiality, capacity [PG]
	Full Idea: Dunamis: faculty, potentiality, capacity
	From: PG (Db (lexicon) [c.1001 BCE], 25)

11321	Eidos: form, idea [PG]
	Full Idea: Eidos: form, idea
	From: PG (Db (lexicon) [c.1001 BCE], 26)
	A reaction: In Plato it is the word best translated as 'Form' (Theory of...); in Aritotle's 'Categories' it designates the species, and in 'Metaphysics' it ends up naming the structural form of the species (and hence the essence) [Wedin p.120]

11322	Elenchos: elenchus, interrogation [PG]
	Full Idea: Elenchos: elenchus, interrogation
	From: PG (Db (lexicon) [c.1001 BCE], 27)

11323	Empeiron: experience [PG]
	Full Idea: Empeiron: experience
	From: PG (Db (lexicon) [c.1001 BCE], 28)

11324	Energeia: employment, actuality, power? [PG]
	Full Idea: Energeia: employment, actuality, power?
	From: PG (Db (lexicon) [c.1001 BCE], 31)

11325	Enkrateia: control [PG]
	Full Idea: Enkrateia: control
	From: PG (Db (lexicon) [c.1001 BCE], 32)
	A reaction: See 'akrasia', of which this is the opposite. The enkratic person is controlled.

11326	Entelecheia: entelechy, having an end [PG]
	Full Idea: Entelecheia: entelechy, having an end
	From: PG (Db (lexicon) [c.1001 BCE], 33)

11327	Epagoge: induction, explanation [PG]
	Full Idea: Epagoge: induction, explanation, leading on
	From: PG (Db (lexicon) [c.1001 BCE], 34)

11328	Episteme: knowledge, understanding [PG]
	Full Idea: Episteme: knowledge, understanding
	From: PG (Db (lexicon) [c.1001 BCE], 35)
	A reaction: Note that 'episteme' can form a plural in Greek, but we can't say 'knowledges', so we have to say 'branches of knowledge', or 'sciences'.

11329	Epithumia: appetite [PG]
	Full Idea: Epithumia: appetite
	From: PG (Db (lexicon) [c.1001 BCE], 36)

11330	Ergon: function [PG]
	Full Idea: Ergon: function, work
	From: PG (Db (lexicon) [c.1001 BCE], 37)

11331	Eristic: polemic, disputation [PG]
	Full Idea: Eristic: polemic, disputation
	From: PG (Db (lexicon) [c.1001 BCE], 38)
	A reaction: This is confrontational argument, rather than the subtle co-operative dialogue of dialectic. British law courts and the House of Commons are founded on eristic, rather than on dialectic. Could there be a dialectical elected assembly?

11332	Eros: love [PG]
	Full Idea: Eros: love, desire
	From: PG (Db (lexicon) [c.1001 BCE], 41)

11333	Eudaimonia: flourishing, happiness, fulfilment [PG]
	Full Idea: Eudaimonia: flourishing, happiness, fulfilment
	From: PG (Db (lexicon) [c.1001 BCE], 42)
	A reaction: Some people defend 'happiness' as the translation, but that seems to me wildly misleading, since eudaimonia is something like life going well, and certainly isn't a psychological state - and definitely not pleasure.

11334	Genos: type, genus [PG]
	Full Idea: Genos: type, genus, kind
	From: PG (Db (lexicon) [c.1001 BCE], 43)

11335	Hexis: state, habit [PG]
	Full Idea: Hexis: state, habit
	From: PG (Db (lexicon) [c.1001 BCE], 44)

11336	Horismos: definition [PG]
	Full Idea: Horismos: definition
	From: PG (Db (lexicon) [c.1001 BCE], 45)

11337	Hule: matter [PG]
	Full Idea: Hule: matter
	From: PG (Db (lexicon) [c.1001 BCE], 46)
	A reaction: The first half of the 'hylomorphism' of Aristotle. See 'morphe'!

11338	Hupokeimenon: subject, underlying thing [cf. Tode ti] [PG]
	Full Idea: Hupokeimenon: subject, underlying thing, substratum [cf. Tode ti]
	From: PG (Db (lexicon) [c.1001 BCE], 47)
	A reaction: Literally 'that which lies under'. Latin version is 'substratum'. In Aristotle it is the problem, of explaining what lies under. It is not the theory that there is some entity called a 'substratum'.

11339	Kalos / kalon: beauty, fineness, nobility [PG]
	Full Idea: Kalos / kalon: beauty, fineness, nobility
	From: PG (Db (lexicon) [c.1001 BCE], 48)
	A reaction: A revealing Greek word, which is not only our rather pure notion of 'beauty', but also seems to mean something like wow!, and (very suggestive, this) applies as much to actions as to objects.

11340	Kath' hauto: in virtue of itself, essentially [PG]
	Full Idea: Kath' hauto: in virtue of itself, essentially
	From: PG (Db (lexicon) [c.1001 BCE], 51)

11341	Kinesis: movement, process [PG]
	Full Idea: Kinesis: movement, process, change
	From: PG (Db (lexicon) [c.1001 BCE], 52)

11342	Kosmos: order, universe [PG]
	Full Idea: Kosmos: order, universe
	From: PG (Db (lexicon) [c.1001 BCE], 53)

11343	Logos: reason, account, word [PG]
	Full Idea: Logos: reason, account, word
	From: PG (Db (lexicon) [c.1001 BCE], 54)

11344	Meson: the mean [PG]
	Full Idea: Meson: the mean
	From: PG (Db (lexicon) [c.1001 BCE], 55)
	A reaction: This is not the 'average', and hence not some theoretical mid-point. I would call it the 'appropriate compromise', remembering that an extreme may be appropriate in certain circumstances.

11345	Metechein: partaking, sharing [PG]
	Full Idea: Metechein: partaking, sharing
	From: PG (Db (lexicon) [c.1001 BCE], 56)
	A reaction: The key word in Plato for the difficult question of the relationships between the Forms and the particulars. The latter 'partake' of the former. Hm. Compare modern 'instantiation', which strikes me as being equally problematic.

11377	Mimesis: imitation, fine art [PG]
	Full Idea: Mimesis: imitation, fine art
	From: PG (Db (lexicon) [c.1001 BCE], 57)

11346	Morphe: form [PG]
	Full Idea: Morphe: form
	From: PG (Db (lexicon) [c.1001 BCE], 58)

11347	Noesis: intellection, rational thought [cf. Dianoia] [PG]
	Full Idea: Noesis: intellection, rational thought [cf. Dianoia]
	From: PG (Db (lexicon) [c.1001 BCE], 59)

11348	Nomos: convention, law, custom [PG]
	Full Idea: Nomos: convention, law, custom
	From: PG (Db (lexicon) [c.1001 BCE], 61)

11349	Nous: intuition, intellect, understanding [PG]
	Full Idea: Nous: intuition, intellect
	From: PG (Db (lexicon) [c.1001 BCE], 62)
	A reaction: There is a condensed discussion of 'nous' in Aristotle's Posterior Analytics B.19

11350	Orexis: desire [PG]
	Full Idea: Orexis: desire
	From: PG (Db (lexicon) [c.1001 BCE], 63)

11351	Ousia: substance, (primary) being, [see 'Prote ousia'] [PG]
	Full Idea: Ousia: substance, (primary) being [see 'Prote ousia']
	From: PG (Db (lexicon) [c.1001 BCE], 64)
	A reaction: It is based on the verb 'to be'. Latin therefore translated it as 'essentia' (esse: to be), and we have ended up translating it as 'essence', but this is wrong! 'Being' is the best translation, and 'substance' is OK. It is the problem, not the answer.

11352	Pathos: emotion, affection, property [PG]
	Full Idea: Pathos: emotion, affection, property
	From: PG (Db (lexicon) [c.1001 BCE], 65)

11353	Phantasia: imagination [PG]
	Full Idea: Phantasia: imagination
	From: PG (Db (lexicon) [c.1001 BCE], 66)

11354	Philia: friendship [PG]
	Full Idea: Philia: friendship
	From: PG (Db (lexicon) [c.1001 BCE], 67)

11355	Philosophia: philosophy, love of wisdom [PG]
	Full Idea: Philosophia: philosophy, love of wisdom
	From: PG (Db (lexicon) [c.1001 BCE], 68)
	A reaction: The point of the word is its claim only to love wisdom, and not actually to be wise.

11356	Phronesis: prudence, practical reason, common sense [PG]
	Full Idea: Phronesis: prudence, practical reason, common sense
	From: PG (Db (lexicon) [c.1001 BCE], 71)
	A reaction: None of the experts use my own translation, which is 'common sense', but that seems to me to perfectly fit all of Aristotle's discussions of the word in 'Ethics'. 'Prudence' seems a daft translation in modern English.

11357	Physis: nature [PG]
	Full Idea: Physis: nature
	From: PG (Db (lexicon) [c.1001 BCE], 72)

11358	Praxis: action, activity [PG]
	Full Idea: Praxis: action, activity
	From: PG (Db (lexicon) [c.1001 BCE], 73)

11359	Prote ousia: primary being [PG]
	Full Idea: Prote ousia: primary being
	From: PG (Db (lexicon) [c.1001 BCE], 74)
	A reaction: The main topic of investigation in Aristotle's 'Metaphysics'. 'Ousia' is the central problem of the text, NOT the answer to the problem.

11360	Psuche: mind, soul, life [PG]
	Full Idea: Psuche: mind, soul, life
	From: PG (Db (lexicon) [c.1001 BCE], 75)
	A reaction: The interesting thing about this is that we have tended to translate it as 'soul', but Aristotle says plants have it, and not merely conscious beings. It is something like the 'form' of a living thing, but then 'form' is a misleading translation too.

11361	Sophia: wisdom [PG]
	Full Idea: Sophia: wisdom
	From: PG (Db (lexicon) [c.1001 BCE], 76)

11362	Sophrosune: moderation, self-control [PG]
	Full Idea: Sophrosune: moderation, self-control
	From: PG (Db (lexicon) [c.1001 BCE], 77)

11363	Stoicheia: elements [PG]
	Full Idea: Stoicheia: elements
	From: PG (Db (lexicon) [c.1001 BCE], 78)

11364	Sullogismos: deduction, syllogism [PG]
	Full Idea: Sullogismos: deduction, syllogism
	From: PG (Db (lexicon) [c.1001 BCE], 81)

11365	Techne: skill, practical knowledge [PG]
	Full Idea: Techne: skill, practical knowledge
	From: PG (Db (lexicon) [c.1001 BCE], 82)

11366	Telos: purpose, end [PG]
	Full Idea: Telos: purpose, end
	From: PG (Db (lexicon) [c.1001 BCE], 83)

11367	Theoria: contemplation [PG]
	Full Idea: Theoria: contemplation
	From: PG (Db (lexicon) [c.1001 BCE], 84)

11368	Theos: god [PG]
	Full Idea: Theos: god
	From: PG (Db (lexicon) [c.1001 BCE], 85)

11369	Ti esti: what-something-is, essence [PG]
	Full Idea: Ti esti: the what-something-is, essence, whatness
	From: PG (Db (lexicon) [c.1001 BCE], 86)

11370	Timoria: vengeance, punishment [PG]
	Full Idea: Timoria: vengeance, punishment
	From: PG (Db (lexicon) [c.1001 BCE], 87)

11371	To ti en einai: essence, what-it-is-to-be [PG]
	Full Idea: To ti en einai: essence, what-it-is-to-be
	From: PG (Db (lexicon) [c.1001 BCE], 88)
	A reaction: This is Aristotle's main term for what we would now call the 'essence'. It is still not a theory of essence, merely an identification of the target. 'Form' is the nearest we get to his actual theory.

11372	To ti estin: essence [PG]
	Full Idea: To ti estin: essence
	From: PG (Db (lexicon) [c.1001 BCE], 91)

11373	Tode ti: this-such, subject of predication [cf. hupokeimenon] [PG]
	Full Idea: Tode ti: this-something, subject of predication, thisness [cf. hupokeimenon]
	From: PG (Db (lexicon) [c.1001 BCE], 92)

1. Philosophy / C. History of Philosophy / 2. Ancient Philosophy / a. Ancient chronology

11461	323 (roughly): Euclid wrote 'Elements', summarising all of geometry [PG]
	Full Idea: Euclid: In around 323 BCE Euclid wrote his 'Elements', summarising all of known geometry.
	From: PG (Db (chronology) [2030])

11390	1000 (roughly): Upanishads written (in Sanskrit); religious and philosophical texts [PG]
	Full Idea: In around 1000 BCE the Upanishads were written, the most philosophical of ancient Hindu texts
	From: PG (Db (chronology) [2030], 0001)

11391	750 (roughly): the Book of Genesis written by Hebrew writers [PG]
	Full Idea: In around 750 BCE the Book of Genesis was written by an anonymous jewish writer
	From: PG (Db (chronology) [2030], 0250)

11392	586: eclipse of the sun on the coast of modern Turkey was predicted by Thales of Miletus [PG]
	Full Idea: In 585 BCE there was an eclipse of the sun, which Thales of Miletus is said to have predicted
	From: PG (Db (chronology) [2030], 0415)

11395	570: Anaximander flourished in Miletus [PG]
	Full Idea: Anaximander: In around 570 BCE the philosopher and astronomer Anaximander flourished in Miletus
	From: PG (Db (chronology) [2030], 0430)

11396	563: the Buddha born in northern India [PG]
	Full Idea: In around 563 BCE Siddhartha Gautama, the Buddha, was born in northern India
	From: PG (Db (chronology) [2030], 0437)

11398	540: Lao Tzu wrote 'Tao Te Ching', the basis of Taoism [PG]
	Full Idea: In around 540 BCE Lao Tzu wrote the 'Tao Te Ching', the basis of Taoism
	From: PG (Db (chronology) [2030], 0460)

11400	529: Pythagoras created his secretive community at Croton in Sicily [PG]
	Full Idea: In around 529 BCE Pythagoras set up a community in Croton, with strict and secret rules and teachings
	From: PG (Db (chronology) [2030], 0471)

11403	500: Heraclitus flourishes at Ephesus, in modern Turkey [PG]
	Full Idea: In around 500 BCE Heraclitus flourished in the city of Ephesus in Ionia
	From: PG (Db (chronology) [2030], 0500)

11404	496: Confucius travels widely, persuading rulers to be more moral [PG]
	Full Idea: In 496 BCE Confucius began a period of wandering, to persuade rulers to be more moral
	From: PG (Db (chronology) [2030], 0504)

11408	472: Empedocles persuades his city (Acragas in Sicily) to become a democracy [PG]
	Full Idea: In 472 BCE Empedocles helped his city of Acragas change to democracy
	From: PG (Db (chronology) [2030], 0528)

11412	450 (roughly): Parmenides and Zeno visit Athens from Italy [PG]
	Full Idea: In around 450 BCE Parmenides and Zeno visited the festival in Athens
	From: PG (Db (chronology) [2030], 0550)

11414	445: Protagoras helps write laws for the new colony of Thurii [PG]
	Full Idea: In 443 BCE Protagoras helped write the laws for the new colony of Thurii
	From: PG (Db (chronology) [2030], 0557)

11417	436 (roughly): Anaxagoras is tried for impiety, and expelled from Athens [PG]
	Full Idea: In about 436 BCE Anaxagoras was tried on a charge of impiety and expelled from Athens
	From: PG (Db (chronology) [2030], 0564)

11421	427: Gorgias visited Athens as ambassador for Leontini [PG]
	Full Idea: In 427 BCE Gorgias of Leontini visited Athens as an ambassador for his city
	From: PG (Db (chronology) [2030], 0573)

11425	399: Socrates executed (with Plato absent through ill health) [PG]
	Full Idea: In 399 BCE Plato was unwell, and was not present at the death of Socrates
	From: PG (Db (chronology) [2030], 0601)

11432	387 (roughly): Plato returned to Athens, and founded the Academy [PG]
	Full Idea: In about 387 BCE Plato returned to Athens and founded his new school at the Academy
	From: PG (Db (chronology) [2030], 0613)

11433	387 (roughly): Aristippus the Elder founder a hedonist school at Cyrene [PG]
	Full Idea: In around 387 BCE a new school was founded at Cyrene by Aristippus the elder
	From: PG (Db (chronology) [2030], 0613)

11440	367: the teenaged Aristotle came to study at the Academy [PG]
	Full Idea: In 367 BCE the seventeen-year-old Aristotle came south to study at the Academy
	From: PG (Db (chronology) [2030], 0633)

11443	360 (roughly): Diogenes of Sinope lives in a barrel in central Athens [PG]
	Full Idea: In around 360 BCE Diogenes of Sinope was living in a barrel in the Agora in Athens
	From: PG (Db (chronology) [2030], 0640)

11445	347: death of Plato [PG]
	Full Idea: In 347 BCE Plato died
	From: PG (Db (chronology) [2030], 0653)

11454	343: Aristotle becomes tutor to 13 year old Alexander (the Great) [PG]
	Full Idea: In 343 BCE at Stagira Aristotle became personal tutor to the thirteen-year-old Alexander (the Great)
	From: PG (Db (chronology) [2030], 0657)

11456	335: Arisotle founded his school at the Lyceum in Athens [PG]
	Full Idea: In 335 BCE Aristotle founded the Lyceum in Athens
	From: PG (Db (chronology) [2030], 0665)

11459	330 (roughly): Chuang Tzu wrote his Taoist book [PG]
	Full Idea: In around 330 BCE Chuang Tzu wrote a key work in the Taoist tradition
	From: PG (Db (chronology) [2030], 0670)

11465	322: Aristotle retired to Chalcis, and died there [PG]
	Full Idea: In 322 BCE Aristotle retired to Chalcis in Euboea, where he died
	From: PG (Db (chronology) [2030], 0678)

11468	307 (roughly): Epicurus founded his school at the Garden in Athens [PG]
	Full Idea: In about 307 BCE Epicurus founded his school at the Garden in Athens
	From: PG (Db (chronology) [2030], 0693)

11470	301 (roughly): Zeno of Citium founded Stoicism at the Stoa Poikile in Athens [PG]
	Full Idea: In about 301 BCE the Stoic school was founded by Zeno of Citium in the Stoa Poikile in Athens
	From: PG (Db (chronology) [2030], 0699)

11483	261: Cleanthes replaced Zeno as head of the Stoa [PG]
	Full Idea: In 261 BCE Cleanthes took over from Zeno as head of the Stoa.
	From: PG (Db (chronology) [2030], 0739)

11486	229 (roughly): Chrysippus replaced Cleanthes has head of the Stoa [PG]
	Full Idea: In about 229 BCE Chrysippus took over from Cleanthes as the head of the Stoic school
	From: PG (Db (chronology) [2030], 0771)

11492	157 (roughly): Carneades became head of the Academy [PG]
	Full Idea: In around 157 BCE Carneades took over as head of the Academy from Hegesinus
	From: PG (Db (chronology) [2030], 0843)

11509	85: most philosophical activity moves to Alexandria [PG]
	Full Idea: In around 85 BCE Athens went into philosophical decline, and leadership moved to Alexandria
	From: PG (Db (chronology) [2030], 0915)

11513	78: Cicero visited the stoic school on Rhodes [PG]
	Full Idea: In around 78 BCE Cicero visited the school of Posidonius in Rhodes.
	From: PG (Db (chronology) [2030], 0922)

11516	60 (roughly): Lucretius wrote his Latin poem on epicureanism [PG]
	Full Idea: In around 60 BCE Lucretius wrote his Latin poem on Epicureanism
	From: PG (Db (chronology) [2030], 0940)

11528	65: Seneca forced to commit suicide by Nero [PG]
	Full Idea: In 65 CE Seneca was forced to commit suicide by the Emperor Nero.
	From: PG (Db (chronology) [2030], 1065)

11531	80: the discourses of the stoic Epictetus are written down [PG]
	Full Idea: In around 80 CE the 'Discourses' of the freed slave Epictetus were written down in Rome.
	From: PG (Db (chronology) [2030], 1080)

11535	170 (roughly): Marcus Aurelius wrote his private stoic meditations [PG]
	Full Idea: In around 170 CE the Emperor Marcus Aurelius wrote his 'Meditations' for private reading.
	From: PG (Db (chronology) [2030], 1170)

11537	-200 (roughly): Sextus Empiricus wrote a series of books on scepticism [PG]
	Full Idea: In around 200 CE Sextus Empiricus wrote a series of books (which survive) defending scepticism
	From: PG (Db (chronology) [2030], 1200)

11541	263: Porphyry began to study with Plotinus in Rome [PG]
	Full Idea: In 263 CE Porphyry joined Plotinus' classes in Rome
	From: PG (Db (chronology) [2030], 1263)

11545	310: Christianity became the official religion of the Roman empire [PG]
	Full Idea: In 310 CE Christianity became the official religion of the Roman Empire
	From: PG (Db (chronology) [2030], 1310)

11549	387: Ambrose converts Augustine to Christianity [PG]
	Full Idea: In 387 CE Augustine converted to Christianity in Milan, guided by St Ambrose
	From: PG (Db (chronology) [2030], 1387)

11555	523: Boethius imprisoned at Pavia, and begins to write [PG]
	Full Idea: In 523 CE Boethius was imprisoned in exile at Pavia, and wrote 'Consolations of Philosophy'
	From: PG (Db (chronology) [2030], 1523)

11557	529: the emperor Justinian closes all the philosophy schools in Athens [PG]
	Full Idea: In 529 CE the Emperor Justinian closed all the philosophy schools in Athens
	From: PG (Db (chronology) [2030], 1529)

1. Philosophy / C. History of Philosophy / 3. Earlier European Philosophy / a. Earlier European chronology

11558	622 (roughly): Mohammed writes the Koran [PG]
	Full Idea: Mohammed: In about 622 CE Muhammed wrote the basic text of Islam, the Koran.
	From: PG (Db (chronology) [2030], 1622)

11559	642: Arabs close the philosophy schools in Alexandria [PG]
	Full Idea: In 642 CE Alexandria was captured by the Arabs, and the philosophy schools were closed
	From: PG (Db (chronology) [2030], 1642)

11560	910 (roughly): Al-Farabi wrote Arabic commentaries on Aristotle [PG]
	Full Idea: Alfarabi: In around 910 CE Al-Farabi explained and expanded Aristotle for the Islamic world.
	From: PG (Db (chronology) [2030], 1910)

11562	1015 (roughly): Ibn Sina (Avicenna) writes a book on Aristotle [PG]
	Full Idea: In around 1015 Avicenna produced his Platonised version of Aristotle in 'The Healing'
	From: PG (Db (chronology) [2030], 2015)

11564	1090: Anselm publishes his proof of the existence of God [PG]
	Full Idea: Anselm: In about 1090 St Anselm of Canterbury publishes his Ontological Proof of God's existence
	From: PG (Db (chronology) [2030], 2090)

11566	1115: Abelard is the chief logic teacher in Paris [PG]
	Full Idea: In around 1115 Abelard became established as the chief logic teacher in Paris
	From: PG (Db (chronology) [2030], 2115)

11573	1166: Ibn Rushd (Averroes) wrote extensive commentaries on Aristotle [PG]
	Full Idea: In around 1166 Averroes (Ibn Rushd), in Seville, wrote extensive commentaries on Aristotle
	From: PG (Db (chronology) [2030], 2166)

11581	1266: Aquinas began writing 'Summa Theologica' [PG]
	Full Idea: In 1266 Aquinas began writing his great theological work, the 'Summa Theologica'
	From: PG (Db (chronology) [2030], 2266)

11586	1280: after his death, the teaching of Aquinas becomes official Dominican doctrine [PG]
	Full Idea: In around 1280 Aquinas's teaching became the official theology of the Dominican order
	From: PG (Db (chronology) [2030], 2280)

11591	1328: William of Ockham decides the Pope is a heretic, and moves to Munich [PG]
	Full Idea: In 1328 William of Ockham decided the Pope was a heretic, and moved to Munich
	From: PG (Db (chronology) [2030], 2328)

17916	1347: the Church persecutes philosophical heresies [PG]
	Full Idea: In 1347 the Church began extensive persecution of unorthodox philosophical thought
	From: PG (Db (chronology) [2030], 2347)

11593	1470: Marsilio Ficino founds a Platonic Academy in Florence [PG]
	Full Idea: In around 1470 Marsilio Ficino founded a Platonic Academy in Florence
	From: PG (Db (chronology) [2030], 2470)

11596	1513: Machiavelli wrote 'The Prince' [PG]
	Full Idea: In 1513 Machiavelli wrote 'The Prince', a tough view of political theory.
	From: PG (Db (chronology) [2030], 2513)

11599	1543: Copernicus publishes his heliocentric view of the solar system [PG]
	Full Idea: In 1543 Nicholas Copernicus, a Polish monk, publishes his new theory of the solar system.
	From: PG (Db (chronology) [2030], 2543)

11601	1580: Montaigne publishes his essays [PG]
	Full Idea: In 1580 Montaigne published a volume of his 'Essays'
	From: PG (Db (chronology) [2030], 2580)

11607	1600: Giordano Bruno was burned at the stake in Rome [PG]
	Full Idea: In 1600 Giordano Bruno was burnt at the stake in Rome, largely for endorsing Copernicus
	From: PG (Db (chronology) [2030], 2600)

1. Philosophy / C. History of Philosophy / 4. Later European Philosophy / a. Later European chronology

11613	1619: Descartes's famous day of meditation inside a stove [PG]
	Full Idea: In 1619 Descartes had a famous day of meditation in a heated stove at Ulm
	From: PG (Db (chronology) [2030], 2619)

11614	1620: Bacon publishes 'Novum Organum' [PG]
	Full Idea: Francis Bacon: In 1620 Bacon published his 'Novum Organon', urging the rise of experimental science
	From: PG (Db (chronology) [2030], 2620)

11619	1633: Galileo convicted of heresy by the Inquisition [PG]
	Full Idea: In 1633 Galileo was condemned to life emprisonment for contradicting church teachings.
	From: PG (Db (chronology) [2030], 2633)

11623	1641: Descartes publishes his 'Meditations' [PG]
	Full Idea: In 1641 Descartes published his well-known 'Meditations', complete with Objections and Replies
	From: PG (Db (chronology) [2030], 2641)

11626	1650: death of Descartes, in Stockholm [PG]
	Full Idea: In 1650 Descartes died in Stockholm, after stressful work for Queen Christina
	From: PG (Db (chronology) [2030], 2650)

11627	1651: Hobbes publishes 'Leviathan' [PG]
	Full Idea: In 1651 Hobbes published his great work on politics and contract morality, 'Leviathan'
	From: PG (Db (chronology) [2030], 2651)

11633	1662: the Port Royal Logic is published [PG]
	Full Idea: Antoine Arnauld: In 1662 Arnauld and Nicole published their famous text, the 'Port-Royal Logic'
	From: PG (Db (chronology) [2030], 2662)

11634	1665: Spinoza writes his 'Ethics' [PG]
	Full Idea: In 1665 the first draft of Spinoza's 'Ethics', his major work, was finished, and published posthumously
	From: PG (Db (chronology) [2030], 2665)

11643	1676: Leibniz settled as librarian to the Duke of Brunswick [PG]
	Full Idea: In 1676 Leibniz became librarian to the Duke of Brunswick, staying for the rest of his life
	From: PG (Db (chronology) [2030], 2676)

11649	1687: Newton publishes his 'Principia Mathematica' [PG]
	Full Idea: In 1687 Newton published his 'Principia', containing his theory of gravity.
	From: PG (Db (chronology) [2030], 2687)

11652	1690: Locke publishes his 'Essay' [PG]
	Full Idea: In 1690 Locke published his 'Essay', his major work on empiricism
	From: PG (Db (chronology) [2030], 2690)

11654	1697: Bayle publishes his 'Dictionary' [PG]
	Full Idea: Pierre Bayle: In about 1697 Pierre Bayle published his 'Historical and Critical Dictionary'
	From: PG (Db (chronology) [2030], 2697)

11659	1713: Berkeley publishes his 'Three Dialogues' [PG]
	Full Idea: In 1713 Berkeley published a popular account of his empiricist idealism in 'Three Dialogues'
	From: PG (Db (chronology) [2030], 2713)

11666	1734: Voltaire publishes his 'Philosophical Letters' [PG]
	Full Idea: Francois-Marie Voltaire: In 1734 Voltaire's 'Lettres Philosophiques' praised liberalism and empiricism
	From: PG (Db (chronology) [2030], 2734)

11667	1739: Hume publishes his 'Treatise' [PG]
	Full Idea: In 1739 Hume returned to Edinburgh and published his 'Treatise', but it sold very few copies
	From: PG (Db (chronology) [2030], 2739)

11675	1762: Rousseau publishes his 'Social Contract' [PG]
	Full Idea: In 1762 Rousseau published his 'Social Contract', basing politics on the popular will
	From: PG (Db (chronology) [2030], 2762)

11682	1781: Kant publishes his 'Critique of Pure Reason' [PG]
	Full Idea: In 1781 Kant published his first great work, the 'Critique of Pure Reason'
	From: PG (Db (chronology) [2030], 2781)

11683	1785: Reid publishes his essays defending common sense [PG]
	Full Idea: In 1785 Thomas Reid, based in Glasgow, published essays defending common sense.
	From: PG (Db (chronology) [2030], 2785)

11687	1798: the French Revolution [PG]
	Full Idea: In 1789 the French Revolution gave strong impetus to the anti-rational 'Romantic' movement
	From: PG (Db (chronology) [2030], 2789)

11694	1807: Hegel publishes his 'Phenomenology of Spirit' [PG]
	Full Idea: In 1807 Hegel published his first major work, the 'Phenomenology of Spirit'
	From: PG (Db (chronology) [2030], 2807)

11701	1818: Schopenhauer publishes his 'World as Will and Idea' [PG]
	Full Idea: In 1818 Schopenhauer published 'The World as Will and Idea', his major work
	From: PG (Db (chronology) [2030], 2818)

11710	1840: Kierkegaard is writing extensively in Copenhagen [PG]
	Full Idea: In around 1840 Kierkegaard lived a quiet life as a writer in Copenhagen
	From: PG (Db (chronology) [2030], 2840)

11713	1843: Mill publishes his 'System of Logic' [PG]
	Full Idea: In 1843 Mill published his 'System of Logic'
	From: PG (Db (chronology) [2030], 2843)

11715	1848: Marx and Engels publis the Communist Manifesto [PG]
	Full Idea: Karl Marx: In 1848 Marx and Engels published their 'Communist Manifesto'
	From: PG (Db (chronology) [2030], 2848)

11717	1859: Darwin publishes his 'Origin of the Species' [PG]
	Full Idea: Charles Darwin: In 1859 Charles Darwin published his theory of natural selection in 'Origin of the Species'.
	From: PG (Db (chronology) [2030], 2859)

11721	1861: Mill publishes 'Utilitarianism' [PG]
	Full Idea: In 1861 Mill published his book 'Utilitarianism'
	From: PG (Db (chronology) [2030], 2861)

11724	1867: Marx begins publishing 'Das Kapital' [PG]
	Full Idea: Karl Marx: In 1867 Karl Marx began publishing his political work 'Das Kapital'
	From: PG (Db (chronology) [2030], 2867)

1. Philosophy / C. History of Philosophy / 5. Modern Philosophy / a. Modern philosophy chronology

11733	1879: Peirce taught for five years at Johns Hopkins University [PG]
	Full Idea: In 1879 Peirce began five years of teaching at Johns Hopkins University
	From: PG (Db (chronology) [2030], 2879)

17907	1879: Frege invents predicate logic [PG]
	Full Idea: In 1879 Frege published his 'Concept Script', which created predicate logic
	From: PG (Db (chronology) [2030], 2879)

17909	1892: Frege's essay 'Sense and Reference' [PG]
	Full Idea: In 1892 Frege published his famous essay 'Sense and Reference' (Sinn und Bedeutung)
	From: PG (Db (chronology) [2030], 2882)

17908	1884: Frege publishes his 'Foundations of Arithmetic' [PG]
	Full Idea: In 1884 Frege published his 'Foundations of Arithmetic', the beginning of logicism
	From: PG (Db (chronology) [2030], 2884)

11735	1885: Nietzsche completed 'Thus Spake Zarathustra' [PG]
	Full Idea: In about 1885 Nietzsche completed his book 'Also Sprach Zarathustra'
	From: PG (Db (chronology) [2030], 2885)

17911	1888: Dedekind publishes axioms for arithmetic [PG]
	Full Idea: In 1888 Dedekind created simple axioms for arithmetic (the Peano Axioms)
	From: PG (Db (chronology) [2030], 2888)

11740	1890: James published 'Principles of Psychology' [PG]
	Full Idea: In 1890 James published his 'Principles of Psychology'
	From: PG (Db (chronology) [2030], 2890)

11742	1895 (roughly): Freud developed theories of the unconscious [PG]
	Full Idea: In around 1895 Sigmund Freud developed his theories of the unconscious mind
	From: PG (Db (chronology) [2030], 2895)

11745	1900: Husserl began developing Phenomenology [PG]
	Full Idea: In 1900 Edmund Husserl began presenting his new philosophy of Phenomenology
	From: PG (Db (chronology) [2030], 2900)

11746	1903: Moore published 'Principia Ethica' [PG]
	Full Idea: In 1903 G.E. Moore published his 'Principia Ethica', attacking naturalistic ethics.
	From: PG (Db (chronology) [2030], 2903)

11747	1904: Dewey became professor at Columbia University [PG]
	Full Idea: In 1904 Dewey moved to Columbia University in New York.
	From: PG (Db (chronology) [2030], 2904)

17910	1908: Zermelo publishes axioms for set theory [PG]
	Full Idea: In 1908 Zermelo published an axiomatisation of the new set theory
	From: PG (Db (chronology) [2030], 2908)

11752	1910: Russell and Whitehead begin publishing 'Principia Mathematica' [PG]
	Full Idea: In 1910 Russell began publication of 'Principia Mathematica', with Whitehead
	From: PG (Db (chronology) [2030], 2910)

11756	1912: Russell meets Wittgenstein in Cambridge [PG]
	Full Idea: In 1912 Russell met Wittgenstein at Cambridge
	From: PG (Db (chronology) [2030], 2912)

11762	1921: Wittgenstein's 'Tractatus' published [PG]
	Full Idea: In 1921 Wittgenstein's 'Tractatus' was published
	From: PG (Db (chronology) [2030], 2921)

11765	1927: Heidegger's 'Being and Time' published [PG]
	Full Idea: In 1927 Heidegger's major work, 'Being and Time', was published
	From: PG (Db (chronology) [2030], 2927)

11768	1930: Frank Ramsey dies at 27 [PG]
	Full Idea: In 1930 Frank Ramsey died at the age of 27.
	From: PG (Db (chronology) [2030], 2930)

11770	1931: Gödel's Incompleteness Theorems [PG]
	Full Idea: Kurt Gödel: In 1931 the mathematician Kurt Gödel publishes his Incompleteness Theorems.
	From: PG (Db (chronology) [2030], 2931)

11773	1933: Tarski's theory of truth [PG]
	Full Idea: Alfred Tarski: In 1933 Alfred Tarski wrote a famous paper presenting a semantic theory of truth.
	From: PG (Db (chronology) [2030], 2933)

11783	1942: Camus published 'The Myth of Sisyphus' [PG]
	Full Idea: In 1942 Camus published 'The Myth of Sisyphus', exploring suicide and the absurd
	From: PG (Db (chronology) [2030], 2942)

11784	1943: Sartre's 'Being and Nothingness' [PG]
	Full Idea: In 1943 Jean-Paul Sartre published his major work, 'Being and Nothingness'
	From: PG (Db (chronology) [2030], 2943)

11787	1945: Merleau-Ponty's 'Phenomenology of Perception' [PG]
	Full Idea: Maurice Merleau-Ponty: In 1945 Maurice Merleau-Pont published 'The Phenomenology of Perception'
	From: PG (Db (chronology) [2030], 2945)

17918	1947: Carnap published 'Meaning and Necessity' [PG]
	Full Idea: In 1947 Carnap published 'Meaning and Necessity'
	From: PG (Db (chronology) [2030], 2947)

11794	1950: Quine's essay 'Two Dogmas of Empiricism' [PG]
	Full Idea: In 1950 Willard Quine published 'Two Dogmas of Empiricism', attacking analytic truth
	From: PG (Db (chronology) [2030], 2950)

17917	1953: Wittgenstein's 'Philosophical Investigations' [PG]
	Full Idea: In 1953 Wittgenstein's posthumous work 'Philosophical Investigations' is published
	From: PG (Db (chronology) [2030], 2953)

17919	1956: Place proposed mind-brain identity [PG]
	Full Idea: In 1956 U.T. Place proposed that the mind is identical to the brain
	From: PG (Db (chronology) [2030], 2956)

11804	1962: Kuhn's 'Structure of Scientific Revolutions' [PG]
	Full Idea: In 1962 Thomas Kuhn's 'Structure of Scientific Revolutions' questioned the authority of science
	From: PG (Db (chronology) [2030], 2962)

17921	1967: Putnam proposed functionalism of the mind [PG]
	Full Idea: In 1967 Putname proposed the functionalist view of the mind
	From: PG (Db (chronology) [2030], 2967)

11808	1971: Rawls's 'A Theory of Justice' [PG]
	Full Idea: In 1971 John Rawls published his famous defence of liberalism in 'A Theory of Justice'
	From: PG (Db (chronology) [2030], 2971)

11810	1972: Kripke publishes 'Naming and Necessity' [PG]
	Full Idea: In 1972 Saul Kripke's 'Naming and Necessity' revised theories about language and reality
	From: PG (Db (chronology) [2030], 2972)

11813	1975: Singer publishes 'Animal Rights' [PG]
	Full Idea: Peter Singer: In 1975 Peter Singer's 'Animal Rights' turned the attention of philosophers to applied ethics.
	From: PG (Db (chronology) [2030], 2975)

17920	1975: Putnam published his Twin Earth example [PG]
	Full Idea: In 1975 Putnam published 'The Meaning of 'Meaning'', containing his Twin Earth example
	From: PG (Db (chronology) [2030], 2975)

11820	1986: David Lewis publishes 'On the Plurality of Worlds' [PG]
	Full Idea: In 1986 David Lewis published 'On the Plurality of Worlds', about possible worlds.
	From: PG (Db (chronology) [2030], 2986)

1. Philosophy / D. Nature of Philosophy / 7. Despair over Philosophy

6602	Philosophy is like a statue which is worshipped but never advances [Bacon]
	Full Idea: Philosophy and the intellectual sciences stand like statues, worshipped and celebrated, but not moved or advanced.
	From: Francis Bacon (Preface to Great Instauration (Renewal) [1620], Vol.4.14), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5
	A reaction: Still the view of most scientists, I suspect. Personally I disagree, because I think philosophy has made enormous advances, in accurate analysis of arguments. The trouble is there is so much of it that it is hard to discern, and we don't live long enough.

1. Philosophy / E. Nature of Metaphysics / 1. Nature of Metaphysics

12124	Metaphysics is the best knowledge, because it is the simplest [Bacon]
	Full Idea: That knowledge is worthiest which is charged with least multiplicity, which appeareth to be metaphysic
	From: Francis Bacon (The Advancement of Learning [1605], II.VII.6)
	A reaction: A surprising view, coming from the father of modern science, but essentially correct. Obviously metaphysics aspires to avoid multiplicity, but it is riddled not only with complexity in its researches, but massive uncertainties.

1. Philosophy / E. Nature of Metaphysics / 4. Metaphysics as Science

12123	Natural history supports physical knowledge, which supports metaphysical knowledge [Bacon]
	Full Idea: Knowledges are as pyramides, whereof history is the basis. So of natural philosophy, the basis is natural history, the stage next the basis is physic; the stage next the vertical point is metaphysic.
	From: Francis Bacon (The Advancement of Learning [1605], II.VII.6)
	A reaction: The father of modern science keeps a place for metaphysics, as the most abstract level above the physical sciences. I would say he is right. It leads to my own slogan: science is the servant of philosophy.

1. Philosophy / E. Nature of Metaphysics / 5. Metaphysics beyond Science

12119	Physics studies transitory matter; metaphysics what is abstracted and necessary [Bacon]
	Full Idea: Physic should contemplate that which is inherent in matter, and therefore transitory; and metaphysic that which is abstracted and fixed
	From: Francis Bacon (The Advancement of Learning [1605], II.VII.3)
	A reaction: He cites the ancients for this view, with which he agrees. One could do worse than hang onto metaphysics as the study of necessities, but must then face the attacks of the Quineans - that knowledge of necessities is beyond us.

12120	Physics is of material and efficient causes, metaphysics of formal and final causes [Bacon]
	Full Idea: Physic inquireth and handleth the material and efficient causes; and metaphysic handleth the formal and final causes.
	From: Francis Bacon (The Advancement of Learning [1605], II.VII.3)
	A reaction: Compare Idea 12119. This divides up Aristotle's famous Four Causes (or Explanations), outlined in 'Physics' II.3. The concept of 'matter', and the nature of 'cause' seem to me to fall with the purview of metaphysics. Interesting, though.

1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis

4465	Note that "is" can assert existence, or predication, or identity, or classification [PG]
	Full Idea: There are four uses of the word "is" in English: as existence ('he is at home'), as predication ('he is tall'), as identity ('he is the man I saw'), and as classification ('he is British').
	From: PG (Db (ideas) [2031])
	A reaction: This seems a nice instance of the sort of point made by analytical philosophy, which can lead to horrible confusion in other breeds of philosophy when it is overlooked.

2. Reason / A. Nature of Reason / 6. Coherence

10237	Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro]
	Full Idea: I take 'coherence' to be a primitive, intuitive notion, not reduced to something formal, and so I do not venture a rigorous definition of it.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
	A reaction: I agree strongly with this. Best to talk of 'the space of reasons', or some such. Rationality extends far beyond what can be formally defined. Coherence is the last court of appeal in rational thought.

2. Reason / D. Definition / 7. Contextual Definition

10204	An 'implicit definition' gives a direct description of the relations of an entity [Shapiro]
	Full Idea: An 'implicit definition' characterizes a structure or class of structures by giving a direct description of the relations that hold among the places of the structure.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
	A reaction: This might also be thought of as a 'functional definition', since it seems to say what the structure or entity does, rather than give the intrinsic characteristics that make its relations and actions possible.

2. Reason / F. Fallacies / 1. Fallacy

4686	Fallacies are errors in reasoning, 'formal' if a clear rule is breached, and 'informal' if more general [PG]
	Full Idea: Fallacies are errors in reasoning, labelled as 'formal' if a clear rule has been breached, and 'informal' if some less precise error has been made.
	From: PG (Db (ideas) [2031])
	A reaction: Presumably there can be a grey area between the two.

2. Reason / F. Fallacies / 3. Question Begging

7415	Question-begging assumes the proposition which is being challenged [PG]
	Full Idea: To beg the question is to take for granted in your argument that very proposition which is being challenged
	From: PG (Db (ideas) [2031])
	A reaction: An undoubted fallacy, and a simple failure to engage in the rational enterprise. I suppose one might give a reason for something, under the mistaken apprehension that it didn't beg the question; analysis of logical form is then needed.

2. Reason / F. Fallacies / 6. Fallacy of Division

7414	What is true of a set is also true of its members [PG]
	Full Idea: The fallacy of division is the claim that what is true of a set must therefore be true of its members.
	From: PG (Db (ideas) [2031])
	A reaction: Clearly a fallacy, but if you only accept sets which are rational, then there is always a reason why a particular is a member of a set, and you can infer facts about particulars from the nature of the set

2. Reason / F. Fallacies / 7. Ad Hominem

6696	The Ad Hominem Fallacy criticises the speaker rather than the argument [PG]
	Full Idea: The Ad Hominem Fallacy is to criticise the person proposing an argument rather than the argument itself, as when you say "You would say that", or "Your behaviour contradicts what you just said".
	From: PG (Db (ideas) [2031])
	A reaction: Nietzsche is very keen on ad hominem arguments, and cheerfully insults great philosophers, but then he doesn't believe there is such a thing as 'pure argument', and he is a relativist.

3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth

13634	Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro]
	Full Idea: In a sense, satisfaction is the notion of 'truth in a model', and (as Hodes 1984 elegantly puts it) 'truth in a model' is a model of 'truth'.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1)
	A reaction: So we can say that Tarski doesn't offer a definition of truth itself, but replaces it with a 'model' of truth.

3. Truth / H. Deflationary Truth / 3. Minimalist Truth

4687	Minimal theories of truth avoid ontological commitment to such things as 'facts' or 'reality' [PG]
	Full Idea: Minimalist theories of truth are those which involve minimum ontological commitment, avoiding references to 'reality' or 'facts' or 'what works', preferring to refer to formal relationships within language.
	From: PG (Db (ideas) [2031])
	A reaction: Personally I am suspicious of minimal theories, which seem to be designed by and for anti-realists. They seem too focused on language, when animals can obviously formulate correct propositions. I'm quite happy with the 'facts', even if that is vague.

4. Formal Logic / A. Syllogistic Logic / 1. Aristotelian Logic

13643	Aristotelian logic is complete [Shapiro]
	Full Idea: Aristotelian logic is complete.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.5)
	A reaction: [He cites Corcoran 1972]

4. Formal Logic / D. Modal Logic ML / 1. Modal Logic

10206	Modal operators are usually treated as quantifiers [Shapiro]
	Full Idea: It is common now, and throughout the history of philosophy, to interpret modal operators as quantifiers. This is an analysis of modality in terms of ontology.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)

4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set

13651	A set is 'transitive' if contains every member of each of its members [Shapiro]
	Full Idea: If, for every b∈d, a∈b entails that a∈d, the d is said to be 'transitive'. In other words, d is transitive if it contains every member of each of its members.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.2)
	A reaction: The alternative would be that the members of the set are subsets, but the members of those subsets are not themselves members of the higher-level set.

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX

13647	Choice is essential for proving downward Löwenheim-Skolem [Shapiro]
	Full Idea: The axiom of choice is essential for proving the downward Löwenheim-Skolem Theorem.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)

10208	Axiom of Choice: some function has a value for every set in a given set [Shapiro]
	Full Idea: One version of the Axiom of Choice says that for every set A of nonempty sets, there is a function whose domain is A and whose value, for every a ∈ A, is a member of a.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 1)

10252	The Axiom of Choice seems to license an infinite amount of choosing [Shapiro]
	Full Idea: If the Axiom of Choice says we can choose one member from each of a set of non-empty sets and put the chosen elements together in a set, this licenses the constructor to do an infinite amount of choosing.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.3)
	A reaction: This is one reason why the Axiom was originally controversial, and still is for many philosophers.

10301	The axiom of choice is controversial, but it could be replaced [Shapiro]
	Full Idea: The axiom of choice has a troubled history, but is now standard in mathematics. It could be replaced with a principle of comprehension for functions), or one could omit the variables ranging over functions.
	From: Stewart Shapiro (Higher-Order Logic [2001], n 3)

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing

13631	Are sets part of logic, or part of mathematics? [Shapiro]
	Full Idea: Is there a notion of set in the jurisdiction of logic, or does it belong to mathematics proper?
	From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
	A reaction: It immediately strikes me that they might be neither. I don't see that relations between well-defined groups of things must involve number, and I don't see that mapping the relations must intrinsically involve logical consequence or inference.

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets

13654	It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro]
	Full Idea: In set theory it is central to the iterative conception that the membership relation is well-founded, ...which means there are no infinite descending chains from any relation.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.4)

13640	Russell's paradox shows that there are classes which are not iterative sets [Shapiro]
	Full Idea: The argument behind Russell's paradox shows that in set theory there are logical sets (i.e. classes) that are not iterative sets.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.3)
	A reaction: In his preface, Shapiro expresses doubts about the idea of a 'logical set'. Hence the theorists like the iterative hierarchy because it is well-founded and under control, not because it is comprehensive in scope. See all of pp.19-20.

13666	Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro]
	Full Idea: Iterative sets do not exhibit a Boolean structure, because the complement of an iterative set is not itself an iterative set.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1)

4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets

13653	'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro]
	Full Idea: A 'well-ordering' of a set X is an irreflexive, transitive, and binary relation on X in which every non-empty subset of X has a least element.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.3)
	A reaction: So there is a beginning, an ongoing sequence, and no retracing of steps.

4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory

10207	Anti-realists reject set theory [Shapiro]
	Full Idea: Anti-realists reject set theory.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
	A reaction: That is, anti-realists about mathematical objects. I would have thought that one could accept an account of sets as (say) fictions, which provided interesting models of mathematics etc.

5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic

13627	There is no 'correct' logic for natural languages [Shapiro]
	Full Idea: There is no question of finding the 'correct' or 'true' logic underlying a part of natural language.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
	A reaction: One needs the context of Shapiro's defence of second-order logic to see his reasons for this. Call me romantic, but I retain faith that there is one true logic. The Kennedy Assassination problem - can't see the truth because drowning in evidence.

13642	Logic is the ideal for learning new propositions on the basis of others [Shapiro]
	Full Idea: A logic can be seen as the ideal of what may be called 'relative justification', the process of coming to know some propositions on the basis of others.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.3.1)
	A reaction: This seems to be the modern idea of logic, as opposed to identification of a set of 'logical truths' from which eternal necessities (such as mathematics) can be derived. 'Know' implies that they are true - which conclusions may not be.

5. Theory of Logic / A. Overview of Logic / 2. History of Logic

13668	Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro]
	Full Idea: Bernays (1918) formulated and proved the completeness of propositional logic, the first precise solution as part of the Hilbert programme.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2.1)

13669	Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro]
	Full Idea: In 1910 Weyl observed that set theory seemed to presuppose natural numbers, and he regarded numbers as more fundamental than sets, as did Fraenkel. Dedekind had developed set theory independently, and used it to formulate numbers.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2.2)

13667	Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro]
	Full Idea: Skolem and Gödel were the main proponents of first-order languages. The higher-order language 'opposition' was championed by Zermelo, Hilbert, and Bernays.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2)

5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic

13662	First-order logic was an afterthought in the development of modern logic [Shapiro]
	Full Idea: Almost all the systems developed in the first part of the twentieth century are higher-order; first-order logic was an afterthought.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1)

13624	The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro]
	Full Idea: The 'triumph' of first-order logic may be related to the remnants of failed foundationalist programmes early this century - logicism and the Hilbert programme.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
	A reaction: Being complete must also be one of its attractions, and Quine seems to like it because of its minimal ontological commitment.

13660	Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro]
	Full Idea: Tharp (1975) suggested that compactness, semantic effectiveness, and the Löwenheim-Skolem properties are consequences of features one would want a logic to have.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5)
	A reaction: I like this proposal, though Shapiro is strongly against. We keep extending our logic so that we can prove new things, but why should we assume that we can prove everything? That's just what Gödel suggests that we should give up on.

13673	The notion of finitude is actually built into first-order languages [Shapiro]
	Full Idea: The notion of finitude is explicitly 'built in' to the systems of first-order languages in one way or another.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1)
	A reaction: Personally I am inclined to think that they are none the worse for that. No one had even thought of all these lovely infinities before 1870, and now we are supposed to change our logic (our actual logic!) to accommodate them. Cf quantum logic.

10588	First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]
	Full Idea: Early study of first-order logic revealed a number of important features. Gödel showed that there is a complete, sound and effective deductive system. It follows that it is Compact, and there are also the downward and upward Löwenheim-Skolem Theorems.
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.1)

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic

15944	Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine]
	Full Idea: Shapiro preferred second-order logic to set theory because second-order logic refers only to the relations and operations in a domain, and not to the other things that set-theory brings with it - other domains, higher-order relations, and so forth.
	From: report of Stewart Shapiro (Foundations without Foundationalism [1991]) by Shaughan Lavine - Understanding the Infinite VII.4

13629	Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro]
	Full Idea: Three systems of semantics for second-order languages: 'standard semantics' (variables cover all relations and functions), 'Henkin semantics' (relations and functions are a subclass) and 'first-order semantics' (many-sorted domains for variable-types).
	From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
	A reaction: [my summary]

13650	Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro]
	Full Idea: In 'Henkin' semantics, in a given model the relation variables range over a fixed collection of relations D on the domain, and the function variables range over a collection of functions F on the domain.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 3.3)

10298	Some say that second-order logic is mathematics, not logic [Shapiro]
	Full Idea: Some authors argue that second-order logic (with standard semantics) is not logic at all, but is a rather obscure form of mathematics.
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.4)

13645	In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro]
	Full Idea: In the standard semantics of second-order logic, by fixing a domain one thereby fixes the range of both the first-order variables and the second-order variables. There is no further 'interpreting' to be done.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 3.3)
	A reaction: This contrasts with 'Henkin' semantics (Idea 13650), or first-order semantics, which involve more than one domain of quantification.

13649	Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro]
	Full Idea: The counterparts of Completeness, Compactness and the Löwenheim-Skolem theorems all fail for second-order languages with standard semantics, but hold for Henkin or first-order semantics. Hence such logics are much like first-order logic.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)
	A reaction: Shapiro votes for the standard semantics, because he wants the greater expressive power, especially for the characterization of infinite structures.

10299	If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]
	Full Idea: If the goal of logical study is to present a canon of inference, a calculus which codifies correct inference patterns, then second-order logic is a non-starter.
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.4)
	A reaction: This seems to be because it is not 'complete'. However, moves like plural quantification seem aimed at capturing ordinary language inferences, so the difficulty is only that there isn't a precise 'calculus'.

5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence

10300	Logical consequence can be defined in terms of the logical terminology [Shapiro]
	Full Idea: Informally, logical consequence is sometimes defined in terms of the meanings of a certain collection of terms, the so-called 'logical terminology'.
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.4)
	A reaction: This seems to be a compositional account, where we build a full account from an account of the atomic bits, perhaps presented as truth-tables.

5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence

10259	The two standard explanations of consequence are semantic (in models) and deductive [Shapiro]
	Full Idea: The two best historical explanations of consequence are the semantic (model-theoretic), and the deductive versions.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.2)
	A reaction: Shapiro points out the fictionalists are in trouble here, because the first involves commitment to sets, and the second to the existence of deductions.

5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=

13626	Semantic consequence is ineffective in second-order logic [Shapiro]
	Full Idea: It follows from Gödel's incompleteness theorem that the semantic consequence relation of second-order logic is not effective. For example, the set of logical truths of any second-order logic is not recursively enumerable. It is not even arithmetic.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
	A reaction: I don't fully understand this, but it sounds rather major, and a good reason to avoid second-order logic (despite Shapiro's proselytising). See Peter Smith on 'effectively enumerable'.

13637	If a logic is incomplete, its semantic consequence relation is not effective [Shapiro]
	Full Idea: Second-order logic is inherently incomplete, so its semantic consequence relation is not effective.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1)

5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens

10257	Intuitionism only sanctions modus ponens if all three components are proved [Shapiro]
	Full Idea: In some intuitionist semantics modus ponens is not sanctioned. At any given time there is likely to be a conditional such that it and its antecedent have been proved, but nobody has bothered to prove the consequent.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.7)
	A reaction: [He cites Heyting] This is a bit baffling. In what sense can 'it' (i.e. the conditional implication) have been 'proved' if the consequent doesn't immediately follow? Proving both propositions seems to make the conditional redundant.

5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic

10253	Either logic determines objects, or objects determine logic, or they are separate [Shapiro]
	Full Idea: Ontology does not depend on language and logic if either one has the objects determining the logic, or the objects are independent of the logic.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.4)
	A reaction: I favour the first option. I think we should seek an account of how logic grows from our understanding of the physical world. If this cannot be established, I shall invent a new Mad Logic, and use it for all my future reasoning, with (I trust) impunity.

5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle

10251	The law of excluded middle might be seen as a principle of omniscience [Shapiro]
	Full Idea: The law of excluded middle might be seen as a principle of omniscience.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.3)
	A reaction: [E.Bishop 1967 is cited] Put that way, you can see why a lot of people (such as intuitionists in mathematics) might begin to doubt it.

8729	Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
	Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2)
	A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle.

5. Theory of Logic / E. Structures of Logic / 1. Logical Form

13632	Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro]
	Full Idea: It is sometimes difficult to find a formula that is a suitable counterpart of a particular sentence of natural language, and there is no acclaimed criterion for what counts as a good, or even acceptable, 'translation'.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1)

5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives

10212	Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro]
	Full Idea: To some extent, every truth-functional connective differs from its counterpart in ordinary language. Classical conjunction, for example, is timeless, whereas the word 'and' often is not. 'Socrates runs and Socrates stops' cannot be reversed.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 3)
	A reaction: Shapiro suggests two interpretations: either the classical connectives are revealing the deeper structure of ordinary language, or else they are a simplification of it.

5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic

10209	A function is just an arbitrary correspondence between collections [Shapiro]
	Full Idea: The modern extensional notion of function is just an arbitrary correspondence between collections.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 1)
	A reaction: Shapiro links this with the idea that a set is just an arbitrary collection. These minimalist concepts seem like a reaction to a general failure to come up with a more useful and common sense definition.

5. Theory of Logic / G. Quantification / 4. Substitutional Quantification

13674	We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro]
	Full Idea: The main role of substitutional semantics is to reduce ontology. As an alternative to model-theoretic semantics for formal languages, the idea is to replace the 'satisfaction' relation of formulas (by objects) with the 'truth' of sentences (using terms).
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4)
	A reaction: I find this very appealing, and Ruth Barcan Marcus is the person to look at. My intuition is that logic should have no ontology at all, as it is just about how inference works, not about how things are. Shapiro offers a compromise.

5. Theory of Logic / G. Quantification / 5. Second-Order Quantification

10290	Second-order variables also range over properties, sets, relations or functions [Shapiro]
	Full Idea: Second-order variables can range over properties, sets, or relations on the items in the domain-of-discourse, or over functions from the domain itself.
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.1)

5. Theory of Logic / G. Quantification / 6. Plural Quantification

10268	Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro]
	Full Idea: Maybe plural quantifiers should themselves be understood in terms of classes (or sets).
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.4)
	A reaction: [Shapiro credits Resnik for this criticism]

5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction

13633	'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro]
	Full Idea: The 'satisfaction' relation may be thought of as a function from models, assignments, and formulas to the truth values {true,false}.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1)
	A reaction: This at least makes clear that satisfaction is not the same as truth. Now you have to understand how Tarski can define truth in terms of satisfaction.

10235	A sentence is 'satisfiable' if it has a model [Shapiro]
	Full Idea: Normally, to say that a sentence Φ is 'satisfiable' is to say that there exists a model of Φ.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
	A reaction: Nothing is said about whether the model is impressive, or founded on good axioms. Tarski builds his account of truth from this initial notion of satisfaction.

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models

10239	The central notion of model theory is the relation of 'satisfaction' [Shapiro]
	Full Idea: The central notion of model theory is the relation of 'satisfaction', sometimes called 'truth in a model'.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9)

10240	Model theory deals with relations, reference and extensions [Shapiro]
	Full Idea: Model theory determines only the relations between truth conditions, the reference of singular terms, the extensions of predicates, and the extensions of the logical terminology.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9)

13644	Semantics for models uses set-theory [Shapiro]
	Full Idea: Typically, model-theoretic semantics is formulated in set theory.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.5.1)

5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms

10214	Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro]
	Full Idea: No object-language theory determines its ontology by itself. The best possible is that all models are isomorphic, in which case the ontology is determined 'up to isomorphism', but only if the domain is finite, or it is stronger than first-order.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 2.5)
	A reaction: This seems highly significant when ontological claims are being made, and is good support for Shapiro's claim that the structures matter, not the objects. There is a parallel in Tarksi's notion of truth-in-all-models. [The Skolem Paradox is the problem]

10238	The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro]
	Full Idea: Set theorists often point out that the set-theoretical hierarchy contains as many isomorphism types as possible; that is the point of the theory.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
	A reaction: Hence there are a huge number of models for any theory, which are then reduced to the one we want at the level of isomorphism.

13670	Categoricity can't be reached in a first-order language [Shapiro]
	Full Idea: Categoricity cannot be attained in a first-order language.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.3)

13636	An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
	Full Idea: An axiomatization is 'categorical' if all its models are isomorphic to one another; ..hence it has 'essentially only one' interpretation [Veblen 1904].
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1)

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems

10590	Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
	Full Idea: Upward Löwenheim-Skolem: if a set of first-order formulas is satisfied by a domain of at least the natural numbers, then it is satisfied by a model of at least some infinite cardinal.
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.1)

10296	The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
	Full Idea: Both of the Löwenheim-Skolem Theorems fail for second-order languages with a standard semantics
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.3.2)

13658	Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]
	Full Idea: A language has the Downward Löwenheim-Skolem property if each satisfiable countable set of sentences has a model whose domain is at most countable.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5)
	A reaction: This means you can't employ an infinite model to represent a fact about a countable set.

13659	Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]
	Full Idea: A language has the Upward Löwenheim-Skolem property if for each set of sentences whose model has an infinite domain, then it has a model at least as big as each infinite cardinal.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5)
	A reaction: This means you can't have a countable model to represent a fact about infinite sets.

10297	The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
	Full Idea: The Löwenheim-Skolem theorem is usually taken as a sort of defect (often thought to be inevitable) of the first-order logic.
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.4)
	A reaction: [He is quoting Wang 1974 p.154]

13648	The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
	Full Idea: The Löwenheim-Skolem theorems mean that no first-order theory with an infinite model is categorical. If Γ has an infinite model, then it has a model of every infinite cardinality. So first-order languages cannot characterize infinite structures.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)
	A reaction: So much of the debate about different logics hinges on characterizing 'infinite structures' - whatever they are! Shapiro is a leading structuralist in mathematics, so he wants second-order logic to help with his project.

13675	Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro]
	Full Idea: The Upward Löwenheim-Skolem theorem fails (trivially) with substitutional semantics. If there are only countably many terms of the language, then there are no uncountable substitution models.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4)
	A reaction: Better and better. See Idea 13674. Why postulate more objects than you can possibly name? I'm even suspicious of all real numbers, because you can't properly define them in finite terms. Shapiro objects that the uncountable can't be characterized.

10292	Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
	Full Idea: Downward Löwenheim-Skolem: a finite or denumerable set of first-order formulas that is satisfied by a model whose domain is infinite is satisfied in a model whose domain is the natural numbers
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.1)

10234	Any theory with an infinite model has a model of every infinite cardinality [Shapiro]
	Full Idea: The Löwenheim-Skolem theorems (which apply to first-order formal theories) show that any theory with an infinite model has a model of every infinite cardinality.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
	A reaction: This aspect of the theorems is the Skolem Paradox. Shapiro argues that in first-order this infinity of models for arithmetic must be accepted, but he defends second-order model theory, where 'standard' models can be selected.

5. Theory of Logic / K. Features of Logics / 3. Soundness

13635	'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro]
	Full Idea: A logic is 'weakly sound' if every theorem is a logical truth, and 'strongly sound', or simply 'sound', if every deduction from Γ is a semantic consequence of Γ. Soundness indicates that the deductive system is faithful to the semantics.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1)
	A reaction: Similarly, 'weakly complete' is when every logical truth is a theorem.

5. Theory of Logic / K. Features of Logics / 4. Completeness

13628	We can live well without completeness in logic [Shapiro]
	Full Idea: We can live without completeness in logic, and live well.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
	A reaction: This is the kind of heady suggestion that American philosophers love to make. Sounds OK to me, though. Our ability to draw good inferences should be expected to outrun our ability to actually prove them. Completeness is for wimps.

5. Theory of Logic / K. Features of Logics / 6. Compactness

13630	Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]
	Full Idea: It is sometimes said that non-compactness is a defect of second-order logic, but it is a consequence of a crucial strength - its ability to give categorical characterisations of infinite structures.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
	A reaction: The dispute between fans of first- and second-order may hinge on their attitude to the infinite. I note that Skolem, who was not keen on the infinite, stuck to first-order. Should we launch a new Skolemite Crusade?

13646	Compactness is derived from soundness and completeness [Shapiro]
	Full Idea: Compactness is a corollary of soundness and completeness. If Γ is not satisfiable, then, by completeness, Γ is not consistent. But the deductions contain only finite premises. So a finite subset shows the inconsistency.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)
	A reaction: [this is abbreviated, but a proof of compactness] Since all worthwhile logics are sound, this effectively means that completeness entails compactness.

5. Theory of Logic / K. Features of Logics / 9. Expressibility

13661	A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro]
	Full Idea: A logical language is 'semantically effective' if the collection of logically true sentences is a recursively enumerable set of strings.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5)

5. Theory of Logic / L. Paradox / 1. Paradox

6516	Monty Hall Dilemma: do you abandon your preference after Monty eliminates one of the rivals? [PG]
	Full Idea: The Monty Hall Dilemma: Three boxes, one with a big prize; pick one to open. Monty Hall then opens one of the other two, which is empty. You may, if you wish, switch from your box to the other unopened box. Should you?
	From: PG (Db (ideas) [2031])
	A reaction: The other two boxes, as a pair, are more likely contain the prize than your box. Monty Hall has eliminated one of them for you, so you should choose the other one. Your intuition that the two remaining boxes are equal is incorrect!

6. Mathematics / A. Nature of Mathematics / 1. Mathematics

10201	Virtually all of mathematics can be modeled in set theory [Shapiro]
	Full Idea: It is well known that virtually every field of mathematics can be reduced to, or modelled in, set theory.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
	A reaction: The word 'virtually' is tantalising. The fact that something can be 'modeled' in set theory doesn't mean it IS set theory. Most weather can be modeled in a computer.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number

13641	Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro]
	Full Idea: 'Definitions' of integers as pairs of naturals, rationals as pairs of integers, reals as Cauchy sequences of rationals, and complex numbers as pairs of reals are reductive foundations of various fields.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.1)
	A reaction: On p.30 (bottom) Shapiro objects that in the process of reduction the numbers acquire properties they didn't have before.

8763	The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
	Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2)
	A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers

13676	Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro]
	Full Idea: The main problem of characterizing the natural numbers is to state, somehow, that 0,1,2,.... are all the numbers that there are. We have seen that this can be accomplished with a higher-order language, but not in a first-order language.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4)

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers

13677	Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
	Full Idea: By convention, the natural numbers are the finite ordinals, the integers are certain equivalence classes of pairs of finite ordinals, etc.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.3)

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers

10213	Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]
	Full Idea: Real numbers are either Cauchy sequences of rational numbers (interpreted as pairs of integers), or else real numbers can be thought of as Dedekind cuts, certain sets of rational numbers. So π is a Dedekind cut, or an equivalence class of sequences.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 2.5)
	A reaction: This question is parallel to the question of whether natural numbers are Zermelo sets or Von Neumann sets. The famous problem is that there seems no way of deciding. Hence, for Shapiro, we are looking at models, not actual objects.

18243	Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro]
	Full Idea: There is no more to understanding the real-number structure than knowing how to use the language of analysis. .. One learns the axioms of the implicit definition. ...These determine the realtionships between real numbers.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9)
	A reaction: This, of course, is the structuralist view of such things, which isn't really interested in the intrinsic nature of anything, but only in its relations. The slogan that 'meaning is use' seems to be in the background.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy

18249	Cauchy gave a formal definition of a converging sequence. [Shapiro]
	Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4)
	A reaction: The sequence is 'Cauchy' if N exists.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts

18245	Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
	Full Idea: A Dedekind Cut is a division of rationals into two set (A1,A2) where every member of A1 is less than every member of A2. If n is the largest A1 or the smallest A2, the cut is produced by n. Some cuts aren't produced by rationals.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.4)

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis

13652	The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
	Full Idea: The 'continuum' is the cardinality of the powerset of a denumerably infinite set.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.2)

6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics

10236	There is no grounding for mathematics that is more secure than mathematics [Shapiro]
	Full Idea: We cannot ground mathematics in any domain or theory that is more secure than mathematics itself.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
	A reaction: This pronouncement comes after a hundred years of hard work, notably by Gödel, so we'd better believe it. It might explain why Putnam rejects the idea that mathematics needs 'foundations'. Personally I'm prepare to found it in countable objects.

8764	Categories are the best foundation for mathematics [Shapiro]
	Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7)
	A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties.

6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics

10256	For intuitionists, proof is inherently informal [Shapiro]
	Full Idea: For intuitionists, proof is inherently informal.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.7)
	A reaction: This thought is quite appealing, so I may have to take intuitionism more seriously. It connects with my view of coherence, which I take to be a notion far too complex for precise definition. However, we don't want 'proof' to just mean 'persuasive'.

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic

13657	First-order arithmetic can't even represent basic number theory [Shapiro]
	Full Idea: Few theorists consider first-order arithmetic to be an adequate representation of even basic number theory.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 5 n28)
	A reaction: This will be because of Idea 13656. Even 'basic' number theory will include all sorts of vast infinities, and that seems to be where the trouble is.

10202	Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
	Full Idea: The natural-number structure is a pattern common to any system of objects that has a distinguished initial object and a successor relation that satisfies the induction principle
	From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
	A reaction: If you started your number system with 5, and successors were only odd numbers, something would have gone wrong, so a bit more seems to be needed. How do we decided whether the initial object is 0, 1 or 2?

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order

10294	Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro]
	Full Idea: Full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory.
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.1)
	A reaction: [he credits Cowles for this remark] Having an unworkable model theory sounds pretty serious to me, as I'm not inclined to be interested in languages which don't produce models of some sort. Surely models are the whole point?

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic

10205	Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro]
	Full Idea: Originally, the focus of geometry was space - matter and extension - and the subject matter of arithmetic was quantity. Geometry concerned the continuous, whereas arithmetic concerned the discrete. Mathematics left these roots in the nineteenth century.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
	A reaction: Mathematicians can do what they like, but I don't think philosophers of mathematics should lose sight of these two roots. It would be odd if the true nature of mathematics had nothing whatever to do with its origin.

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers

8762	Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
	Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2)
	A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them.

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory

13656	Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
	Full Idea: There are sets of natural numbers definable in set-theory but not in arithmetic.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.3.3)

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory

10222	Mathematical foundations may not be sets; categories are a popular rival [Shapiro]
	Full Idea: Foundationalists (e.g. Quine and Lewis) have shown that mathematics can be rendered in theories other than the iterative hierarchy of sets. A dedicated contingent hold that the category of categories is the proper foundation (e.g. Lawvere).
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3)
	A reaction: I like the sound of that. The categories are presumably concepts that generate sets. Tricky territory, with Frege's disaster as a horrible warning to be careful.

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

10218	Baseball positions and chess pieces depend entirely on context [Shapiro]
	Full Idea: We cannot imagine a shortstop independent of a baseball infield, or a piece that plays the role of black's queen bishop independent of a chess game.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1)
	A reaction: This is the basic thought that leads to the structuralist view of things. I must be careful because I like structuralism, but I have attacked the functionalist view in many areas, because it neglects the essences of the functioning entities.

10224	The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro]
	Full Idea: The even numbers and the natural numbers greater than 4 both exemplify the natural-number structure. In the former, 6 plays the 3 role, and in the latter 8 plays the 3 role.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.5)
	A reaction: This begins to sound a bit odd. If you count the even numbers, 6 is the third one. I could count pebbles using only evens, but then presumably '6' would just mean '3'; it wouldn't be the actual number 6 acting in a different role, like Laurence Olivier.

10228	Could infinite structures be apprehended by pattern recognition? [Shapiro]
	Full Idea: It is contentious, to say the least, to claim that infinite structures are apprehended by pattern recognition.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.1)
	A reaction: It only seems contentious for completed infinities. The idea that the pattern continues in same way seems (pace Wittgenstein) fairly self-evident, just like an arithmetical series.

10230	The 4-pattern is the structure common to all collections of four objects [Shapiro]
	Full Idea: The 4-pattern is the structure common to all collections of four objects.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.2)
	A reaction: This seems open to Frege's objection, that you can have four disparate abstract concepts, or four spatially scattered items of unknown pattern. It certainly isn't a visual pattern, but then if the only detectable pattern is the fourness, it is circular.

10249	The main mathematical structures are algebraic, ordered, and topological [Shapiro]
	Full Idea: According to Bourbaki, there are three main types of structure: algebraic structures, such as group, ring, field; order structures, such as partial order, linear order, well-order; topological structures, involving limit, neighbour, continuity, and space.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.5)
	A reaction: Bourbaki is mentioned as the main champion of structuralism within mathematics.

10273	Some structures are exemplified by both abstract and concrete [Shapiro]
	Full Idea: Some structures are exemplified by both systems of abstracta and systems of concreta.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.2)
	A reaction: It at least seems plausible that one might try to build a physical structure that modelled arithmetic (an abacus might be an instance), so the parallel is feasible. Then to say that the abstract arose from modelling the physical seems equally plausible.

10276	Mathematical structures are defined by axioms, or in set theory [Shapiro]
	Full Idea: Mathematical structures are characterised axiomatically (as implicit definitions), or they are defined in set theory.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.3)
	A reaction: Presumably earlier mathematicians had neither axiomatised their theories, nor expressed them in set theory, but they still had a good working knowledge of the relationships.

8760	Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
	Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1)
	A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate.

8761	A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
	Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1)
	A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts.

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism

10270	The main versions of structuralism are all definitionally equivalent [Shapiro]
	Full Idea: Ante rem structuralism, eliminative structuralism formulated over a sufficiently large domain of abstract objects, and modal eliminative structuralism are all definitionally equivalent. Neither is to be ontologically preferred, but the first is clearer.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.5)
	A reaction: Since Shapiro's ontology is platonist, I would have thought there were pretty obvious grounds for making a choice between that and eliminativm, even if the grounds are intuitive rather than formal.

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism

10221	Is there is no more to structures than the systems that exemplify them? [Shapiro]
	Full Idea: The 'in re' view of structures is that there is no more to structures than the systems that exemplify them.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3)
	A reaction: I say there is more than just the systems, because we can abstract from them to a common structure, but that doesn't commit us to the existence of such a common structure.

10248	Number statements are generalizations about number sequences, and are bound variables [Shapiro]
	Full Idea: According to 'in re' structuralism, a statement that appears to be about numbers is a disguised generalization about all natural-number sequences; the numbers are bound variables, not singular terms.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.3.4)
	A reaction: Any theory of anything which comes out with the thought that 'really it is a variable, not a ...' has my immediate attention and sympathy.

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism

10220	Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro]
	Full Idea: Because the same structure can be exemplified by more than one system, a structure is a one-over-many.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3)
	A reaction: The phrase 'one-over-many' is a classic Greek hallmark of a universal. Cf. Idea 10217, where Shapiro talks of arriving at structures by abstraction, through focusing and ignoring. This sounds more like a creation than a platonic universal.

10223	There is no 'structure of all structures', just as there is no set of all sets [Shapiro]
	Full Idea: There is no 'structure of all structures', just as there is no set of all sets.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.4)
	A reaction: If one cannot abstract from all the structures to a higher level, why should Shapiro have abstracted from the systems/models to get the over-arching structures?

8703	Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend]
	Full Idea: Shapiro's structuralism champions model theory as the branch of mathematics that best describes mathematics. The essence of mathematical activity is seen as an exercise in comparing mathematical structures to each other.
	From: report of Stewart Shapiro (Philosophy of Mathematics [1997], 4.4) by Michèle Friend - Introducing the Philosophy of Mathematics
	A reaction: Note it 'best describes' it, rather than being foundational. Assessing whether propositional logic is complete is given as an example of model theory. That makes model theory a very high-level activity. Does it capture simple arithmetic?

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique

10274	Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
	Full Idea: According to structuralism, someone who uses small natural numbers in everyday life presupposes an infinite structure. It seems absurd that a child who learns to count his toes applies an infinite structure to reality, and thus presupposes the structure.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.2)
	A reaction: Shapiro says we can meet this objection by thinking of smaller structures embedded in larger ones, with the child knowing the smaller ones.

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism

10200	We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro]
	Full Idea: We must distinguish between 'realism in ontology' - that mathematical objects exist - and 'realism in truth-value', which is suggested by the model-theoretic framework - that each well-formed meaningful sentence is non-vacuously either true or false.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
	A reaction: My inclination is fairly strongly towards realism of the second kind, but not of the first. A view about the notion of a 'truth-maker' might therefore be required. What do the truths refer to? Answer: not objects, but abstractions from objects.

10210	If mathematical objects are accepted, then a number of standard principles will follow [Shapiro]
	Full Idea: One who believes in the independent existence of mathematical objects is likely to accept the law of excluded middle, impredicative definitions, the axiom of choice, extensionality, and arbitrary sets and functions.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 1)
	A reaction: The underlying thought is that since the objects pre-exist, all of the above simply describe the relations between them, rather than having to actually bring the objects into existence. Personally I would seek a middle ground.

10215	Platonists claim we can state the essence of a number without reference to the others [Shapiro]
	Full Idea: The Platonist view may be that one can state the essence of each number, without referring to the other numbers.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1)
	A reaction: Frege certainly talks this way (in his 'borehole' analogy). Fine, we are asked to spell out the essence of some number, without making reference either to any 'units' composing it, or to any other number adjacent to it or composing it. Reals?

10233	Platonism must accept that the Peano Axioms could all be false [Shapiro]
	Full Idea: A traditional Platonist has to face the possibility that all of the Peano Axioms are false.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.7)
	A reaction: This would be because the objects exist independently, and so the Axioms are a mere human attempt at pinning them down. For the Formalist the axioms create the numbers, and so couldn't be false. This makes me, alas, warm to platonism!

6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics

10244	Intuition is an outright hindrance to five-dimensional geometry [Shapiro]
	Full Idea: Even if spatial intuition provides a little help in the heuristics of four-dimensional geometry, intuition is an outright hindrance for five-dimensional geometry and beyond.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.2)
	A reaction: One might respond by saying 'so much the worse for five-dimensional geometry'. One could hardly abolish the subject, though, so the point must be taken.

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism

10280	A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro]
	Full Idea: For each stone, there is at least one pattern such that the stone is a position in that pattern. The stone can be treated in terms of places-are-objects, or places-are-offices, to be filled with objects drawn from another ontology.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.4)
	A reaction: I believe this is the story J.S. Mill had in mind. His view was that the structures move off into abstraction, but it is only at the empirical and physical level that we can possibly learn the structures.

6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism

13664	Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro]
	Full Idea: It is claimed that aiming at a universal language for all contexts, and the thesis that logic does not involve a process of abstraction, separates the logicists from algebraists and mathematicians, and also from modern model theory.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1)
	A reaction: I am intuitively drawn to the idea that logic is essentially the result of a series of abstractions, so this gives me a further reason not to be a logicist. Shapiro cites Goldfarb 1979 and van Heijenoort 1967. Logicists reduce abstraction to logic.

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique

13625	Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
	Full Idea: I extend Quinean holism to logic itself; there is no sharp border between mathematics and logic, especially the logic of mathematics. One cannot expect to do logic without incorporating some mathematics and accepting at least some of its ontology.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
	A reaction: I have strong sales resistance to this proposal. Mathematics may have hijacked logic and warped it for its own evil purposes, but if logic is just the study of inferences then it must be more general than to apply specifically to mathematics.

8744	Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
	Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1)
	A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism.

6. Mathematics / C. Sources of Mathematics / 7. Formalism

8749	Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
	Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names?
	From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1)
	A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up.

8750	Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
	Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2)
	A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare.

8752	Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
	Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2)
	A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right.

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism

10254	Can the ideal constructor also destroy objects? [Shapiro]
	Full Idea: Can we assume that the ideal constructor cannot destroy objects? Presumably the ideal constructor does not have an eraser, and the collection of objects is non-reducing over time.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.5)
	A reaction: A very nice question, which platonists should enjoy.

10255	Presumably nothing can block a possible dynamic operation? [Shapiro]
	Full Idea: Presumably within a dynamic system, once the constructor has an operation available, then no activity can preclude the performance of the operation?
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.5)
	A reaction: There seems to be an interesting assumption in static accounts of mathematics, that all the possible outputs of (say) a function actually exist with a theory. In an actual dynamic account, the constructor may be smitten with lethargy.

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism

8753	Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
	Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1)
	A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them.

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism

8731	Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
	Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1)
	A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football.

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism

13663	Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
	Full Idea: Some authors (Poincaré and Russell, for example) were disposed to reject properties that are not definable, or are definable only impredicatively.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1)
	A reaction: I take Quine to be the culmination of this line of thought, with his general rejection of 'attributes' in logic and in metaphysics.

8730	'Impredicative' definitions refer to the thing being described [Shapiro]
	Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2)
	A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise.

7. Existence / A. Nature of Existence / 1. Nature of Existence

10279	Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro]
	Full Idea: Can we 'discover' whether a deck is really identical with its fifty-two cards, or whether a person is identical with her corresponding time-slices, molecules, or space-time points? This is like Benacerraf's problem about numbers.
	From: Stewart Shapiro (Philosophy of Mathematics [1997])
	A reaction: Shapiro is defending the structuralist view, that each of these is a model of an agreed reality, so we cannot choose a right model if they all satisfy the necessary criteria.

7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete

10227	The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro]
	Full Idea: The epistemic proposals of ontological realists in mathematics (such as Maddy and Resnik) has resulted in the blurring of the abstract/concrete boundary. ...Perhaps the burden is now on defenders of the boundary.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.1)
	A reaction: As Shapiro says, 'a vague boundary is still a boundary', so we need not be mesmerised by borderline cases. I would defend the boundary, with the concrete just being physical. A chair is physical, but our concept of a chair may already be abstract.

10226	Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro]
	Full Idea: Mathematicians use the 'abstract/concrete' label differently, with arithmetic being 'concrete' because it is a single structure (up to isomorphism), while group theory is considered more 'abstract'.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.1 n1)
	A reaction: I would say that it is the normal distinction, but they have moved the significant boundary up several levels in the hierarchy of abstraction.

7. Existence / D. Theories of Reality / 7. Fictionalism

10262	Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro]
	Full Idea: Fictionalism takes an epistemology of the concrete to be more promising than concrete-and-abstract, but fictionalism requires an epistemology of the actual and possible, secured without the benefits of model theory.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.2)
	A reaction: The idea that possibilities (logical, natural and metaphysical) should be understood as features of the concrete world has always struck me as appealing, so I have (unlike Shapiro) no intuitive problems with this proposal.

10277	Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro]
	Full Idea: One result of the structuralist perspective is a healthy blurring of the distinction between mathematical and ordinary objects. ..'According to the structuralist, physical configurations often instantiate mathematical patterns'.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.4)
	A reaction: [The quotation is from Penelope Maddy 1988 p.28] This is probably the main reason why I found structuralism interesting, and began to investigate it.

8. Modes of Existence / B. Properties / 10. Properties as Predicates

13638	Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro]
	Full Idea: Properties are often taken to be intensional; equiangular and equilateral are thought to be different properties of triangles, even though any triangle is equilateral if and only if it is equiangular.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.3)
	A reaction: Many logicians seem to want to treat properties as sets of objects (red being just the set of red things), but this looks like a desperate desire to say everything in first-order logic, where only objects are available to quantify over.

8. Modes of Existence / B. Properties / 11. Properties as Sets

10591	Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro]
	Full Idea: In studying second-order logic one can think of relations and functions as extensional or intensional, or one can leave it open. Little turns on this here, and so words like 'property', 'class', and 'set' are used interchangeably.
	From: Stewart Shapiro (Higher-Order Logic [2001], 2.2.1)
	A reaction: Important. Students of the metaphysics of properties, who arrive with limited experience of logic, are bewildered by this attitude. Note that the metaphysics is left wide open, so never let logicians hijack the metaphysical problem of properties.

8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism

16639	Only individual bodies exist [Bacon]
	Full Idea: Nothing truly exists in nature beyond individual bodies.
	From: Francis Bacon (The New Organon [1620]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 182
	A reaction: [Unusually, Pasnau gives no reference in the text; possibly II:1-2] What this leaves out, from even an auster nominalist ontology, is undifferentiated stuff like water. Even electrons don't seem quite distinct from one another.

9. Objects / A. Existence of Objects / 1. Physical Objects

10272	The notion of 'object' is at least partially structural and mathematical [Shapiro]
	Full Idea: The very notion of 'object' is at least partially structural and mathematical.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.1)
	A reaction: [In the context, Shapiro clearly has physical objects in mind] This view seems to fit with Russell's 'relational' view of the physical world, though Russell rejected structuralism in mathematics. I take abstraction to be part of perception.

9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects

10275	A blurry border is still a border [Shapiro]
	Full Idea: A blurry border is still a border.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.3)
	A reaction: This remark deserves to be quoted in almost every area of philosophy, against those who attack a concept by focusing on its vague edges. Philosophers should focus on central cases, not borderline cases (though the latter may be of interest).

9. Objects / C. Structure of Objects / 2. Hylomorphism / c. Form as causal

16625	In hylomorphism all the explanation of actions is in the form, and the matter doesn't do anything [Bacon]
	Full Idea: Prime, common matter seems to be a kind of accessory and to stand as a substratum, whereas any kind of action seems to be a mere emanation of form. So it is that forms are given all the leading parts.
	From: Francis Bacon (Philosophical Studies 1611-19 [1617], p.206), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 07.2
	A reaction: This is a very striking criticism of hylomorphism. The revolution was simple - that actually matter seems to do all the real work, and the form can take a back seat.

16033	There are only individual bodies containing law-based powers, and the Forms are these laws [Bacon]
	Full Idea: Though nothing exists in nature except individual bodies which exhibit pure individual acts [powers] in accordance with law…It is this law and its clauses which we understand by the term Forms.
	From: Francis Bacon (The New Organon [1620], p.103), quoted by Jan-Erik Jones - Real Essence §3
	A reaction: This isn't far off what Aristotle had in mind, when he talks of forms as being 'principles', though there is more emphasis on mechanisms in the original idea. Note that Bacon takes laws so literally that he refers to their 'clauses'.

10. Modality / A. Necessity / 6. Logical Necessity

10258	Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro]
	Full Idea: For many philosophers the logical notions of possibility and necessity are exceptions to a general scepticism, perhaps because they have been reduced to model theory, via set theory. Thus Φ is logically possible if there is a model that satisfies it.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.1)
	A reaction: Initially this looks a bit feeble, like an empiricist only believing what they actually see right now, but the modern analytical philosophy project seems to be the extension of logical accounts further and further into what we intuit about modality.

10. Modality / B. Possibility / 6. Probability

24054	Everything has a probability, something will happen, and probabilities add up [PG]
	Full Idea: The three Kolgorov axioms of probability: the probability of an event is a non-negative real number; it is certain that one of the 'elementary events' will occur; and the unity of probabilities is the sum of probability of parts ('additivity').
	From: PG (Db (ideas) [2031])
	A reaction: [My attempt to verbalise them; they are normally expressed in terms of set theory]. Got this from a talk handout, and Wikipedia.

10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds

10266	Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro]
	Full Idea: The fact that the 'myth' of possible worlds happens to produce the correct modal logic is itself a phenomenon in need of explanation.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.4)
	A reaction: The claim that it produces 'the' correct modal logic seems to beg a lot of questions, given the profusion of modal systems. This is a problem with any sort of metaphysics which invokes fictionalism - what were those particular fictions responding to?

11. Knowledge Aims / C. Knowing Reality / 1. Perceptual Realism / a. Naïve realism

3875	If reality is just what we perceive, we would have no need for a sixth sense [PG]
	Full Idea: Reality must be more than merely what we perceive, because a sixth sense would enhance our current knowledge, and a seventh, and so on.
	From: PG (Db (ideas) [2031])

12. Knowledge Sources / A. A Priori Knowledge / 5. A Priori Synthetic

3876	If my team is losing 3-1, I have synthetic a priori knowledge that they need two goals for a draw [PG]
	Full Idea: If my football team is losing 3-1, I seem to have synthetic a priori knowledge that they need two goals to achieve a draw
	From: PG (Db (ideas) [2031])

12. Knowledge Sources / B. Perception / 1. Perception

16724	The senses deceive, but also show their own errors [Bacon]
	Full Idea: It is certain that the senses deceive, but they also testify to their own errors.
	From: Francis Bacon (Preface to Great Instauration (Renewal) [1620], p.32), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 22.1
	A reaction: Nice. This is the empiricist view, rather than the rationalist line that reason sorts out the mess created by the senses. Most people know things if you just show them.

12. Knowledge Sources / C. Rationalism / 1. Rationalism

8725	Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
	Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1)
	A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach.

12. Knowledge Sources / D. Empiricism / 1. Empiricism

3648	Empiricists are collecting ants; rationalists are spinning spiders; and bees do both [Bacon]
	Full Idea: Empiricists are like ants; they collect and put to use; but rationalists, like spiders, spin threads out of themselves. (…and bees follow the middle way, of collecting material and transforming it).
	From: Francis Bacon (Cogitata et Visa [1607])
	A reaction: Nice (and so concisely expressed). Bees seem to be just more intelligent and energetic empiricists.

12121	We don't assume there is no land, because we can only see sea [Bacon]
	Full Idea: They are ill discoverers that think there is no land, when they can see nothing but sea.
	From: Francis Bacon (The Advancement of Learning [1605], II.VII.5)
	A reaction: Just the sort of pithy remark for which Bacon is famous. It is an obvious point, but a nice corrective to anyone who wants to apply empirical principles in a rather gormless way.

14. Science / A. Basis of Science / 3. Experiment

12117	Science moves up and down between inventions of causes, and experiments [Bacon]
	Full Idea: All true and fruitful natural philosophy hath a double scale or ladder, ascendent and descendent, ascending from experiments to the invention of causes, and descending from causes to the invention of new experiments.
	From: Francis Bacon (The Advancement of Learning [1605], II.VII.1)
	A reaction: After several hundred years, I doubt whether anyone can come up with a better account of scientific method than Bacon's.

6603	Nature is revealed when we put it under pressure rather than observe it [Bacon]
	Full Idea: The secrets of nature reveal themselves more readily under the vexations of art than when they go their own way.
	From: Francis Bacon (Preface to Great Instauration (Renewal) [1620], Vol.4.95), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5
	A reaction: This is a splendid slogan for the dawn of the age of science, and pinpoints the reason why we have advanced so much further than the Greeks. You can, of course, overdo the 'vexations of art'. It also justifies the critical approach to philosophy.

14. Science / B. Scientific Theories / 2. Aim of Science

21950	Science must clear away the idols of the mind if they are ever going to find the truth [Bacon]
	Full Idea: We must clear away the idols and false notions which are now in possession of the human understanding, and have taken deep root therein, and so beset men's minds that truth can hardly find an entrance.
	From: Francis Bacon (The New Organon [1620], 38), quoted by Mark Wrathall - Heidegger: how to read 2
	A reaction: [He goes on to list the types of idol]

14. Science / B. Scientific Theories / 5. Commensurability

12127	Many different theories will fit the observed facts [Bacon]
	Full Idea: The ordinary face and view of experience is many times satisfied by several theories and philosophies.
	From: Francis Bacon (The Advancement of Learning [1605], II.VIII.5)
	A reaction: He gives as his example that the Copernican system and the Ptolemaic system both seem to satisfy all the facts. He wrote in 1605, just before Galileo's telescope. His point is regularly made in modern discussions. In this case, he was wrong!

15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind

10203	We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro]
	Full Idea: The epistemological account of mathematical structures depends on the size and complexity of the structure, but small, finite structures are apprehended through abstraction via simple pattern recognition.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
	A reaction: Yes! This I take to be the reason why John Stuart Mill was not a fool in his discussion of the pebbles. Successive abstractions (and fictions) will then get you to more complex structures.

15. Nature of Minds / C. Capacities of Minds / 5. Generalisation by mind

12126	People love (unfortunately) extreme generality, rather than particular knowledge [Bacon]
	Full Idea: It is the nature of the mind of man (to the extreme prejudice of knowledge) to delight in the spacious liberty of generalities, as in a champaign region, and not in the inclosures of particularity.
	From: Francis Bacon (The Advancement of Learning [1605], II.VIII.1)
	A reaction: I have to plead guilty to this myself. He may have pinpointed the key motivation behind philosophy. We all want to know things, as Aristotle said, but some of us want the broad brush, and others want the fine detail.

17. Mind and Body / E. Mind as Physical / 7. Anti-Physicalism / b. Multiple realisability

7734	Maybe a mollusc's brain events for pain ARE of the same type (broadly) as a human's [PG]
	Full Idea: To defend type-type identity against the multiple realisability objection, we might say that a molluscs's brain events that register pain ARE of the same type as humans, given that being 'of the same type' is a fairly flexible concept.
	From: PG (Db (ideas) [2031])
	A reaction: But this reduces 'of the same type' to such vagueness that it may become vacuous. You would be left with token-token identity, where the mental event is just identical to some brain event, with its 'type' being irrelevant.

7735	Maybe a frog's brain events for fear are functionally like ours, but not phenomenally [PG]
	Full Idea: To defend type-type identity against the multiple realisability objection, we might (also) say that while a frog's brain events for fear are functionally identical to a human's (it runs away), that doesn't mean they are phenomenally identical.
	From: PG (Db (ideas) [2031])
	A reaction: I take this to be the key reply to the multiple realisability problem. If a frog flees from a loud noise, it is 'frightened' in a functional sense, but that still leaves the question 'What's it like to be a frightened frog?', which may differ from humans.

18. Thought / E. Abstraction / 2. Abstracta by Selection

10229	Simple types can be apprehended through their tokens, via abstraction [Shapiro]
	Full Idea: Some realists argue that simple types can be apprehended through their tokens, via abstraction.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.2)
	A reaction: One might rephrase that to say that types are created by abstraction from tokens (and then preserved in language).

18. Thought / E. Abstraction / 3. Abstracta by Ignoring

9626	A structure is an abstraction, focussing on relationships, and ignoring other features [Shapiro]
	Full Idea: A structure is the abstract form of a system, focussing on the interrelationships among the objects, and ignoring any features of them that do not affect how they relate to other objects in the system.
	From: Stewart Shapiro (Structure and Ontology [1989], 146), quoted by James Robert Brown - Philosophy of Mathematics Ch.4
	A reaction: I find this account very attractive, even though it appeals to supposedly outmoded psychological abstractionism. It seems pretty close to Aristotle's view of things. Shapiro's account must face up to Frege's worries about these matters.

10217	We can apprehend structures by focusing on or ignoring features of patterns [Shapiro]
	Full Idea: One way to apprehend a particular structure is through a process of pattern recognition, or abstraction. One observes systems in a structure, and focuses attention on the relations among the objects - ignoring features irrelevant to their relations.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1)
	A reaction: A lovely statement of the classic Aristotelian abstractionist approach of focusing-and-ignoring. But this is made in 1997, long after Frege and Geach ridiculed it. It just won't go away - not if you want a full and unified account of what is going on.

9554	We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro]
	Full Idea: One can observe a system and focus attention on the relations among the objects - ignoring those features of the objects not relevant to the system. For example, we can understand a baseball defense system by going to several games.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], p.74), quoted by Charles Chihara - A Structural Account of Mathematics
	A reaction: This is Shapiro perpetrating precisely the wicked abstractionism which Frege and Geach claim is ridiculous. Frege objects that abstract concepts then become private, but baseball defences are discussed in national newspapers.

18. Thought / E. Abstraction / 7. Abstracta by Equivalence

10231	Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro]
	Full Idea: Perhaps we can introduce abstract objects by abstraction over an equivalence relation on a base class of entities, just as Frege suggested that 'direction' be obtained from parallel lines. ..Properties must be equinumerous, but need not be individuated.
	From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.5)
	A reaction: [He cites Hale and Wright as the originators of this} It is not entirely clear why this is 'abstraction', rather than just drawing attention to possible groupings of entities.

23. Ethics / E. Utilitarianism / 4. Unfairness

3877	Utilitarianism seems to justify the discreet murder of unhappy people [PG]
	Full Idea: If I discreetly murdered a gloomy and solitary tramp who was upsetting people in my village, if is hard to see how utilitarianism could demonstrate that I had done something wrong.
	From: PG (Db (ideas) [2031])

26. Natural Theory / A. Speculations on Nature / 2. Natural Purpose / c. Purpose denied

12125	Teleological accounts are fine in metaphysics, but they stop us from searching for the causes [Bacon]
	Full Idea: To say 'leaves are for protecting of fruit', or that 'clouds are for watering the earth', is well inquired and collected in metaphysic, but in physic they are impertinent. They are hindrances, and the search of the physical causes hath been neglected.
	From: Francis Bacon (The Advancement of Learning [1605], II.VII.7)
	A reaction: This is the standard rebellion against Aristotle which gave rise to the birth of modern science. The story has been complicated by natural selection, which bestows a sort of purpose on living things. Nowadays we pursue both paths.

26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / a. Greek matter

16624	Stripped and passive matter is just a human invention [Bacon]
	Full Idea: Stripped and passive matter seems nothing more than an invention of the human mind.
	From: Francis Bacon (Philosophical Studies 1611-19 [1617], p.206), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 07.2
	A reaction: Bacon seems to me to get too little credit in the history of philosophy, because he is just seen as a progenitor of science. His modern views predate most radical 17th C thought by 20 years.

26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / a. Scientific essentialism

12118	Essences are part of first philosophy, but as part of nature, not part of logic [Bacon]
	Full Idea: I assign to summary philosophy the operation of essences (as quantity, similitude, diversity, possibility), with this distinction - that they be handled as they have efficacy in nature, and not logically.
	From: Francis Bacon (The Advancement of Learning [1605], II.VII.3)
	A reaction: I take this to be a splendid motto for scientific essentialism, in a climate where modal logicians appear to have taken over the driving seat in our understanding of essences.

27. Natural Reality / G. Biology / 2. Life

6126	Life is Movement, Respiration, Sensation, Nutrition, Excretion, Reproduction, Growth (MRS NERG) [PG]
	Full Idea: The biologists' acronym for the necessary conditions of life is MRS NERG: that is, Movement, Respiration, Sensation, Nutrition, Excretion, Reproduction, Growth.
	From: PG (Db (ideas) [2031])
	A reaction: How strictly necessary are each of these is a point for discussion. A notorious problem case is fire, which (at a stretch) may pass all seven tests.

28. God / A. Divine Nature / 4. Divine Contradictions

3873	An omniscient being couldn't know it was omniscient, as that requires information from beyond its scope of knowledge [PG]
	Full Idea: God seems to be in the paradoxical situation that He may be omniscient, but can never know that He is, because that involves knowing that there is nothing outside his scope of knowledge (e.g. another God)
	From: PG (Db (ideas) [2031])

3874	How could God know there wasn't an unknown force controlling his 'free' will? [PG]
	Full Idea: How could God be certain that he has free will (if He has), if He couldn't be sure that there wasn't an unknown force controlling his will?
	From: PG (Db (ideas) [2031])

28. God / A. Divine Nature / 6. Divine Morality / b. Euthyphro question

7399	Even without religion, there are many guides to morality [Bacon]
	Full Idea: Atheism leaves a man to sense, to philosophy, to natural piety, to laws, to reputation; all which may be guides to an outward moral virtue, though religion were not.
	From: Francis Bacon (17: Of Superstition [1625], p.52)
	A reaction: One might add to Bacon's list 'contracts', or 'rational consistency', or 'self-evident human excellence', or 'natural sympathy'. This is a striking idea, which clearly made churchmen uneasy when atheism began to spread.