46 ideas
9641 | Definitions should be replaceable by primitives, and should not be creative [Brown,JR] |
Full Idea: The standard requirement of definitions involves 'eliminability' (any defined terms must be replaceable by primitives) and 'non-creativity' (proofs of theorems should not depend on the definition). | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 7) | |
A reaction: [He cites Russell and Whitehead as a source for this view] This is the austere view of the mathematician or logician. But almost every abstract concept that we use was actually defined in a creative way. |
9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR] |
Full Idea: The set-theory account of infinity doesn't just say that we can keep on counting, but that the natural numbers are an actual infinite set. This is necessary to make sense of the powerset of ω, as the set of all its subsets, and thus even bigger. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5) | |
A reaction: I don't personally find this to be sufficient reason to commit myself to the existence of actual infinities. In fact I have growing doubts about the whole role of set theory in philosophy of mathematics. Shows how much I know. |
9613 | Naïve set theory assumed that there is a set for every condition [Brown,JR] |
Full Idea: In the early versions of set theory ('naïve' set theory), the axiom of comprehension assumed that for any condition there is a set of objects satisfying that condition (so P(x)↔x∈{x:P(x)}), but this led directly to Russell's Paradox. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
A reaction: How rarely any philosophers state this problem clearly (as Brown does here). This is incredibly important for our understanding of how we classify the world. I'm tempted to just ignore Russell, and treat sets in a natural and sensible way. |
9615 | Nowadays conditions are only defined on existing sets [Brown,JR] |
Full Idea: In current set theory Russell's Paradox is avoided by saying that a condition can only be defined on already existing sets. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
A reaction: A response to Idea 9613. This leaves us with no account of how sets are created, so we have the modern notion that absolutely any grouping of daft things is a perfectly good set. The logicians seem to have hijacked common sense. |
9617 | The 'iterative' view says sets start with the empty set and build up [Brown,JR] |
Full Idea: The modern 'iterative' concept of a set starts with the empty set φ (or unsetted individuals), then uses set-forming operations (characterized by the axioms) to build up ever more complex sets. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
A reaction: The only sets in our system will be those we can construct, rather than anything accepted intuitively. It is more about building an elaborate machine that works than about giving a good model of reality. |
9642 | A flock of birds is not a set, because a set cannot go anywhere [Brown,JR] |
Full Idea: Neither a flock of birds nor a pack of wolves is strictly a set, since a flock can fly south, and a pack can be on the prowl, whereas sets go nowhere and menace no one. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 7) | |
A reaction: To say that the pack menaced you would presumably be to commit the fallacy of composition. Doesn't the number 64 have properties which its set-theoretic elements (whatever we decide they are) will lack? |
9605 | If a proposition is false, then its negation is true [Brown,JR] |
Full Idea: The law of excluded middle says if a proposition is false, then its negation is true | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 1) | |
A reaction: Surely that is the best statement of the law? How do you write that down? ¬(P)→¬P? No, because it is a semantic claim, not a syntactic claim, so a truth table captures it. Semantic claims are bigger than syntactic claims. |
9649 | Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR] |
Full Idea: The three views one could adopt concerning axioms are that they are self-evident truths, or that they are arbitrary stipulations, or that they are fallible attempts to describe how things are. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch.10) | |
A reaction: Presumably modern platonists like the third version, with others choosing the second, and hardly anyone now having the confidence to embrace the first. |
9638 | Berry's Paradox finds a contradiction in the naming of huge numbers [Brown,JR] |
Full Idea: Berry's Paradox refers to 'the least integer not namable in fewer than nineteen syllables' - a paradox because it has just been named in eighteen syllables. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5) | |
A reaction: Apparently George Boolos used this quirky idea as a basis for a new and more streamlined proof of Gödel's Theorem. Don't tell me you don't find that impressive. |
9604 | Mathematics is the only place where we are sure we are right [Brown,JR] |
Full Idea: Mathematics seems to be the one and only place where we humans can be absolutely sure that we got it right. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 1) | |
A reaction: Apart from death and taxes, that is. Personally I am more certain of the keyboard I am typing on than I am of Pythagoras's Theorem, but the experts seem pretty confident about the number stuff. |
9622 | 'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR] |
Full Idea: 'There are two apples' can be recast as 'x is an apple and y is an apple, and x isn't y, and if z is an apple it is the same as x or y', which makes no appeal at all to mathematics. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4) | |
A reaction: He cites this as the basis of Hartry Field's claim that science can be done without numbers. The logic is ∃x∃y∀z(Ax&Ay&(x¬=y)&(Az→z=x∨z=y)). |
9648 | π is a 'transcendental' number, because it is not the solution of an equation [Brown,JR] |
Full Idea: The number π is not only irrational, but it is also (unlike √2) a 'transcendental' number, because it is not the solution of an algebraic equation. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch.10) | |
A reaction: So is that a superficial property, or a profound one? Answers on a post card. |
9621 | Mathematics represents the world through structurally similar models. [Brown,JR] |
Full Idea: Mathematics hooks onto the world by providing representations in the form of structurally similar models. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4) | |
A reaction: This is Brown's conclusion. It needs notions of mapping, one-to-one correspondence, and similarity. I like the idea of a 'model', as used in both logic and mathematics, and children's hobbies. The mind is a model-making machine. |
9646 | There is no limit to how many ways something can be proved in mathematics [Brown,JR] |
Full Idea: I'm tempted to say that mathematics is so rich that there are indefinitely many ways to prove anything - verbal/symbolic derivations and pictures are just two. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 9) | |
A reaction: Brown has been defending pictures as a form of proof. I wonder how long his list would be, if we challenged him to give more details? Some people have very low standards of proof. |
9647 | Computers played an essential role in proving the four-colour theorem of maps [Brown,JR] |
Full Idea: The celebrity of the famous proof in 1976 of the four-colour theorem of maps is that a computer played an essential role in the proof. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch.10) | |
A reaction: The problem concerns the reliability of the computers, but then all the people who check a traditional proof might also be unreliable. Quis custodet custodies? |
9643 | Set theory may represent all of mathematics, without actually being mathematics [Brown,JR] |
Full Idea: Maybe all of mathematics can be represented in set theory, but we should not think that mathematics is set theory. Functions can be represented as order pairs, but perhaps that is not what functions really are. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 7) | |
A reaction: This seems to me to be the correct view of the situation. If 2 is represented as {φ,{φ}}, why is that asymmetrical? The first digit seems to be the senior and original partner, but how could the digits of 2 differ from one another? |
9644 | When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR] |
Full Idea: The basic definition of a graph can be given in set-theoretic terms,...but then what could an unlabelled graph be? | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 7) | |
A reaction: An unlabelled graph will at least need a verbal description for it to have any significance at all. My daily mood-swings look like this.... |
9625 | To see a structure in something, we must already have the idea of the structure [Brown,JR] |
Full Idea: Epistemology is a big worry for structuralists. ..To conjecture that something has a particular structure, we must already have conceived of the idea of the structure itself; we cannot be discovering structures by conjecturing them. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4) | |
A reaction: This has to be a crucial area of discussion. Do we have our heads full of abstract structures before we look out of the window? Externalism about the mind is important here; mind and world are not utterly distinct things. |
9628 | Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR] |
Full Idea: Set theory is at the very heart of mathematics; it may even be all there is to mathematics. The notion of set, however, seems quite contrary to the spirit of structuralism. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4) | |
A reaction: So much the worse for sets, I say. You can, for example, define ordinality in terms of sets, but that is no good if ordinality is basic to the nature of numbers, rather than a later addition. |
9606 | The irrationality of root-2 was achieved by intellect, not experience [Brown,JR] |
Full Idea: We could not discover irrational numbers by physical measurement. The discovery of the irrationality of the square root of two was an intellectual achievement, not at all connected to sense experience. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 1) | |
A reaction: Brown declares himself a platonist, and this is clearly a key argument for him, and rather a good one. Hm. I'll get back to you on this one... |
9612 | There is an infinity of mathematical objects, so they can't be physical [Brown,JR] |
Full Idea: A simple argument makes it clear that all mathematical arguments are abstract: there are infinitely many numbers, but only a finite number of physical entities, so most mathematical objects are non-physical. The best assumption is that they all are. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
A reaction: This, it seems to me, is where constructivists score well (cf. Idea 9608). I don't have an infinity of bricks to build an infinity of houses, but I can imagine that the bricks just keep coming if I need them. Imagination is what is unbounded. |
9610 | Numbers are not abstracted from particulars, because each number is a particular [Brown,JR] |
Full Idea: Numbers are not 'abstract' (in the old sense, of universals abstracted from particulars), since each of the integers is a unique individual, a particular, not a universal. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
A reaction: An interesting observation which I have not seen directly stated before. Compare Idea 645. I suspect that numbers should be thought of as higher-order abstractions, which don't behave like normal universals (i.e. they're not distributed). |
9620 | Empiricists base numbers on objects, Platonists base them on properties [Brown,JR] |
Full Idea: Perhaps, instead of objects, numbers are associated with properties of objects. Basing them on objects is strongly empiricist and uses first-order logic, whereas the latter view is somewhat Platonistic, and uses second-order logic. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4) | |
A reaction: I don't seem to have a view on this. You can count tomatoes, or you can count red objects, or even 'instances of red'. Numbers refer to whatever can be individuated. No individuation, no arithmetic. (It's also Hume v Armstrong on laws on nature). |
9629 | For nomalists there are no numbers, only numerals [Brown,JR] |
Full Idea: For the instinctive nominalist in mathematics, there are no numbers, only numerals. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5) | |
A reaction: Maybe. A numeral is a specific sign, sometimes in a specific natural language, so this seems to miss the fact that cardinality etc are features of reality, not just conventions. |
9639 | Does some mathematics depend entirely on notation? [Brown,JR] |
Full Idea: Are there mathematical properties which can only be discovered using a particular notation? | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 6) | |
A reaction: If so, this would seem to be a serious difficulty for platonists. Brown has just been exploring the mathematical theory of knots. |
9630 | The most brilliant formalist was Hilbert [Brown,JR] |
Full Idea: In mathematics, the most brilliant formalist of all was Hilbert | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5) | |
A reaction: He seems to have developed his fully formalist views later in his career. See Mathematics|Basis of Mathematic|Formalism in our thematic section. Kreisel denies that Hilbert was a true formalist. |
9608 | There are no constructions for many highly desirable results in mathematics [Brown,JR] |
Full Idea: Constuctivists link truth with constructive proof, but necessarily lack constructions for many highly desirable results of classical mathematics, making their account of mathematical truth rather implausible. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
A reaction: The tricky word here is 'desirable', which is an odd criterion for mathematical truth. Nevertheless this sounds like a good objection. How flexible might the concept of a 'construction' be? |
9645 | Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR] |
Full Idea: If we define p as '3 if Goldbach's Conjecture is true' and '5 if Goldbach's Conjecture is false', it seems that p must be a prime number, but, amazingly, constructivists would not accept this without a proof of Goldbach's Conjecture. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 8) | |
A reaction: A very similar argument structure to Schrödinger's Cat. This seems (as Brown implies) to be a devastating knock-down argument, but I'll keep an open mind for now. |
9619 | David's 'Napoleon' is about something concrete and something abstract [Brown,JR] |
Full Idea: David's painting of Napoleon (on a white horse) is a 'picture' of Napoleon, and a 'symbol' of leadership, courage, adventure. It manages to be about something concrete and something abstract. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 3) | |
A reaction: This strikes me as the germ of an extremely important idea - that abstraction is involved in our perception of the concrete, so that they are not two entirely separate realms. Seeing 'as' involves abstraction. |
9611 | 'Abstract' nowadays means outside space and time, not concrete, not physical [Brown,JR] |
Full Idea: The current usage of 'abstract' simply means outside space and time, not concrete, not physical. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
A reaction: This is in contrast to Idea 9609 (the older notion of being abstracted). It seems odd that our ancestors had a theory about where such ideas came from, but modern thinkers have no theory at all. Blame Frege for that. |
9609 | The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars [Brown,JR] |
Full Idea: The older sense of 'abstract' applies to universals, where a universal like 'redness' is abstracted from red particulars; it is the one associated with the many. In mathematics, the notion of 'group' or 'vector space' perhaps fits this pattern. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
A reaction: I am currently investigating whether this 'older' concept is in fact dead. It seems to me that it is needed, as part of cognitive science, and as the crucial link between a materialist metaphysic and the world of ideas. |
9640 | A term can have not only a sense and a reference, but also a 'computational role' [Brown,JR] |
Full Idea: In addition to the sense and reference of term, there is the 'computational' role. The name '2' has a sense (successor of 1) and a reference (the number 2). But the word 'two' has little computational power, Roman 'II' is better, and '2' is a marvel. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 6) | |
A reaction: Very interesting, and the point might transfer to natural languages. Synonymous terms carry with them not just different expressive powers, but the capacity to play different roles (e.g. slang and formal terms, gob and mouth). |
7357 | People who control others with fluent language often end up being hated [Kongzi (Confucius)] |
Full Idea: Of what use is eloquence? He who engages in fluency of words to control men often finds himself hated by them. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], V.5) | |
A reaction: I don't recall Socrates making this very good point to any of the sophists (such as Gorgias). The idea that if you battle or connive your way to dominance over others then you are successful is false. Life is a much longer game than that. |
7358 | All men prefer outward appearance to true excellence [Kongzi (Confucius)] |
Full Idea: I have yet to meet a man as fond of excellence as he is of outward appearances. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], IX.18) | |
A reaction: Interestingly, this cynical view of the love of virtue is put by Plato into the mouths of Glaucon and Adeimantus (in Bk II of 'Republic', e.g. Idea 12), and not into the mouth of Socrates, who goes on to defend the possibility of true virtue. |
7362 | Humans are similar, but social conventions drive us apart (sages and idiots being the exceptions) [Kongzi (Confucius)] |
Full Idea: In our natures we approximate one another; habits put us further and further apart. The only ones who do not change are sages and idiots. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XVII.2) | |
A reaction: I find most of Confucius rather uninteresting, but this is a splendid remark about the influence of social conventions on human nature. Sages can achieve universal morality if they rise above social convention, and seek the true virtues of human nature. |
7360 | Do not do to others what you would not desire yourself [Kongzi (Confucius)] |
Full Idea: Do not do to others what you would not desire yourself. Then you will have no enemies, either in the state or in your home. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XII.2) | |
A reaction: The Golden Rule, but note the second sentence. Logically, it leads to the absurdity of not giving someone an Elvis record for Christmas because you yourself don't like Elvis. Kant (Idea 3733) and Nietzsche (Idea 4560) offer good criticisms. |
7359 | Excess and deficiency are equally at fault [Kongzi (Confucius)] |
Full Idea: Excess and deficiency are equally at fault. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XI.16) | |
A reaction: This is the sort of wisdom we admire in Aristotle (and in any sensible person), but it may also be the deepest motto of conservatism, and it is a long way from romantic philosophy, and the clarion call of Nietzsche to greater excitement in life. |
7363 | The virtues of the best people are humility, maganimity, sincerity, diligence, and graciousness [Kongzi (Confucius)] |
Full Idea: He who in this world can practise five things may indeed be considered Man-at-his-best: humility, maganimity, sincerity, diligence, and graciousness. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XVII.5) | |
A reaction: A very nice list. Who could resist working with a colleague who had such virtues? Who could go wrong if they married a person who had them? I can't think of anything important that is missing. |
7361 | Men of the highest calibre avoid political life completely [Kongzi (Confucius)] |
Full Idea: Men of the highest calibre avoid political life completely. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XIV.37) | |
A reaction: Plato notes that such people tend to avoid political life (and a left sheltering, as if from a wild storm!), but he thinks they should be dragged into the political arena for the common good. Confucius seems to approve of the avoidance. Plato is right. |
23393 | Confucianism assumes that all good developments have happened, and there is only one Way [Norden on Kongzi (Confucius)] |
Full Idea: The two major limitations of Confucianism are that it assumes that all worthwhile cultural, social and ethical innovation has already occurred, and that it does not recognise the plurality of worthwhile ways of life. | |
From: comment on Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE]) by Bryan van Norden - Intro to Classical Chinese Philosophy 3.III | |
A reaction: In modern liberal terms that is about as conservative as it is possible to get. We think of it as the state of mind of an old person who can only long for the way things were when they were young. But 'hold fast to that which is good'! |
21733 | The right-wing conception of freedom is based on the idea of self-ownership [Cohen,GA] |
Full Idea: The right-wing conception of freedom is, I think, founded on the idea that each person is the morally rightful owner of himself, even if existing legal systems do not acknowledge it. Let us call that the 'self-ownership' thesis. | |
From: G.A. Cohen (Are Freedom and Equality Compatible? [1986], 1) | |
A reaction: He cites Nozick as articulating this view. At the end Cohen rejects self-ownership, though he agrees that no one would accept that the state could be the owner of your eyes. Do I own my hair after it is cut? |
21739 | Plenty of people have self-ownership, but still lack autonomy [Cohen,GA] |
Full Idea: Universal self-ownership fails to ensure autonomy, since it tends to produce proletarians, who lack it. | |
From: G.A. Cohen (Are Freedom and Equality Compatible? [1986], 3) | |
A reaction: The implication is that autonomy is not a property of individuals but a social phenomenon. Self-owning people can still be imprisoned. What about autonomy without self-ownership? A bright slave who is given extensive responsibility? |
21736 | It is doubtful whether any private property was originally acquired legitimately [Cohen,GA] |
Full Idea: It is easy to doubt that much actually existing private property was formed in what anyone could think was a legitimating way. | |
From: G.A. Cohen (Are Freedom and Equality Compatible? [1986], 2) | |
A reaction: What if I created an artificial island out of unwanted raw materials? What about the first humans to reach some remote territory? |
21734 | It is plausible that no one has an initial right to own land and natural resources [Cohen,GA] |
Full Idea: One may plausibly say of external things in their initial state, of raw land and natural resources, that no person has a greater right to them than any other does. | |
From: G.A. Cohen (Are Freedom and Equality Compatible? [1986], 1) | |
A reaction: How about if your group has lived on that plot for fifty generations, and some interlopers arrive and claim part of it. No one thought of 'owning' it till the interlopers arrived. Native Americans and Australians. |
21735 | Every thing which is now private started out as unowned [Cohen,GA] |
Full Idea: In the prehistory of anything that is now private property there was at least one moment at which something privately unowned was taken into private ownership. | |
From: G.A. Cohen (Are Freedom and Equality Compatible? [1986], 2) | |
A reaction: He is obviously talking about land and natural resources. Presumably a table which I made and own was always private property, although the land where the trees were grown was not. Though in some communities what I make could be automatically communal. |
9635 | Given atomism at one end, and a finite universe at the other, there are no physical infinities [Brown,JR] |
Full Idea: There seem to be no actual infinites in the physical realm. Given the correctness of atomism, there are no infinitely small things, no infinite divisibility. And General Relativity says that the universe is only finitely large. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5) | |
A reaction: If time was infinite, you could travel round in a circle forever. An atom has size, so it has a left, middle and right to it. Etc. They seem to be physical, so we will count those too. |