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10170
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While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price]
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Full Idea:
While truth can be defined in a relative way, as truth in one particular model, a non-relative notion of truth is implied, as truth in all models.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4)
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A reaction:
[The article is actually discussing arithmetic] This idea strikes me as extremely important. True-in-all-models is usually taken to be tautological, but it does seem to give a more universal notion of truth. See semantic truth, Tarski, Davidson etc etc.
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10175
|
Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price]
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Full Idea:
In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
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A reaction:
It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types.
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10164
|
Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price]
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Full Idea:
A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
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A reaction:
This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'!
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10167
|
Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
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Full Idea:
Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
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A reaction:
In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent.
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10169
|
Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
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Full Idea:
Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4)
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A reaction:
The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight?
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10179
|
There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price]
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Full Idea:
The term 'structure' has two uses in the literature, what can be called 'particular structures' (which are particular relational systems), but also what can be called 'universal structures' - what particular systems share, or what they instantiate.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6)
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A reaction:
This is a very helpful distinction, because it clarifies why (rather to my surprise) some structuralists turn out to be platonists in a new guise. Personal my interest in structuralism has been anti-platonist from the start.
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10182
|
There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price]
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Full Idea:
There are four main variants of structuralism in the philosophy of mathematics - formalist structuralism, relativist structuralism, universalist structuralism (with modal variants), and pattern structuralism.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9)
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A reaction:
I'm not sure where Chihara's later book fits into this, though it is at the nominalist end of the spectrum. Shapiro and Resnik do patterns (the latter more loosely); Hellman does modal universalism; Quine does the relativist version. Dedekind?
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10168
|
Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
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Full Idea:
Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations.
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|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3)
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A reaction:
[very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations.
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10178
|
Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
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Full Idea:
It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true.
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|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
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|
A reaction:
[compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent.
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10177
|
Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
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Full Idea:
Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out.
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|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
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A reaction:
I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra.
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20653
|
Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson]
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Full Idea:
There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles.
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|
From:
report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro'
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|
A reaction:
I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture?
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6248
|
Reason is too slow and doubtful to guide all actions, which need external and moral senses [Hutcheson]
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Full Idea:
We boast of our mighty reason above other animals, but its processes are too slow, too full of doubt, to serve us in every exigency, either for our preservation, without external senses, or to influence our actions for good without the moral sense.
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|
From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §VII.III)
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|
A reaction:
This idea was taken up by Hume, and it must have influence Hume's general scepticism about the importance of reason. What this idea misses is the enormous influence of prior reasoning on our quick decisions.
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6239
|
We dislike a traitor, even if they give us great benefit [Hutcheson]
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|
Full Idea:
Let us consider if a traitor, who would sell his own country to us, may not often be as advantageous to us, as an hero who defends us: and yet we can love the treason, and hate the traitor.
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|
From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §I.VI)
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|
A reaction:
A nice example, which certainly refutes any claim that morality is entirely and directly self-interested. High-minded idealism, though, is not the only alternative explanation. We admire loyalty, but not loyalty to, say, Hitler.
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|
6240
|
The moral sense is not an innate idea, but an ability to approve or disapprove in a disinterested way [Hutcheson]
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Full Idea:
The moral sense is not an innate idea or knowledge, but a determination of our minds to receive the simple ideas of approbation or condemnation, from actions observed, antecedent to any opinions of advantage or loss to redound to ourselves.
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|
From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §I.VIII)
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|
A reaction:
This may claim a pure moral intuition, but it is also close to Kantian universalising of the rules for behaviour. It is also a variation on Descartes' 'natural light' of reason. Of course, if we say the ideas are 'received', where are they received from?
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6242
|
We cannot choose our moral feelings, otherwise bribery could affect them [Hutcheson]
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|
Full Idea:
Neither benevolence nor any other affection or desire can be directly raised by volition; if they could, then we could be bribed into any affection whatsoever toward any object.
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|
From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §II.IV)
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|
A reaction:
Of course, notoriously, the vast mass of people have often been bribed to love a politician, by low taxes, or bread and circuses. Still, you cannot choose to love or admire someone, you just do. Not much free will there.
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6244
|
Human nature seems incapable of universal malice, except what results from self-love [Hutcheson]
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Full Idea:
Human nature seems scarce capable of malicious disinterested hatred, or an ultimate desire of the misery of others, when we imagine them not pernicious to us, or opposite to our interests; ..that is only the effect of self-love, not disinterested malice.
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|
From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §II.VII)
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|
A reaction:
I suppose it is true that even the worst criminals brooding in prison don't wish the entire population of some foreign country to die in pain. Only a very freakish person would wish the human race were extinct. A very nice observation.
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6243
|
As death approaches, why do we still care about family, friends or country? [Hutcheson]
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|
Full Idea:
How comes it that we do not lose, at the approach of death, all concern for our families, friends, or country?
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|
From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §II.V)
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|
A reaction:
A nice question. No doubt some people do cease to care, but on the whole it raises the 'last round' problem in social contract theory, which is why fulfil your part of a bargain if it is too late to receive the repayment afterwards?
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|
6241
|
Contempt of danger is just madness if it is not in some worthy cause [Hutcheson]
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|
Full Idea:
Mere courage, or contempt of danger, if we conceive it to have no regard to the defence of the innocent, or repairing of wrongs or self-interest, would only entitle its possessor to bedlam.
|
|
From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §II.I)
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|
A reaction:
If many criminals would love to rob a bank, but only a few have the nerve to attempt it, we can hardly deny that the latter exhibit a sort of courage. The Greeks say that good sense must be involved, but few of them were so moral about courage.
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6257
|
You can't form moral rules without an end, which needs feelings and a moral sense [Hutcheson]
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|
Full Idea:
What rule of actions can be formed, without relation to some end proposed? Or what end can be proposed, without presupposing instincts, desires, affections, or a moral sense, it will not be easy to explain.
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|
From:
Francis Hutcheson (Treatise 4: The Moral Sense [1728], §IV)
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|
A reaction:
We have no reason to think that 'instincts, desires and affections' will give us the remotest guidance on how to behave morally well (though we would expect them to aid our survival). How could a moral sense give a reason, without spotting a rule?
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|
6245
|
That action is best, which procures the greatest happiness for the greatest number [Hutcheson]
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|
Full Idea:
That action is best, which procures the greatest happiness for the greatest number; and that worst, which, in like manner, occasions misery.
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|
From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §III.VIII)
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|
A reaction:
The first use of a phrase taken up by Bentham. This is not just an anticipation of utilitarianism, it is utilitarianism, with all its commitment to consequentialism (but see Idea 6246), and to the maximising of happiness. It is a brilliant idea.
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|
6251
|
The loss of perfect rights causes misery, but the loss of imperfect rights reduces social good [Hutcheson]
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|
Full Idea:
Perfect rights are necessary to the public good, and it makes those miserable whose rights are thus violated; …imperfect rights tend to the improvement and increase of good in a society, but are not necessary to prevent universal misery.
|
|
From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §VII.VI)
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|
A reaction:
This is a very utilitarian streak in Hutcheson, converting natural law into its tangible outcome in actual happiness or misery. The distinction here is interesting (taken up by Mill), but there is a very blurred borderline.
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6254
|
We are asked to follow God's ends because he is our benefactor, but why must we do that? [Hutcheson]
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Full Idea:
The reasons assigned for actions are such as 'It is the end proposed by the Deity'. But why do we approve concurring with the divine ends? The reason is given 'He is our benefactor', but then, for what reason do we approve concurrence with a benefactor?
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|
From:
Francis Hutcheson (Treatise 4: The Moral Sense [1728], §I)
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|
A reaction:
Characteristic of what MacIntyre calls the 'Enlightenment Project', which is the application of Cartesian scepticism to proving the foundations of morals. Proof beyond proof is continually demanded. If you could meet God, you would obey without question.
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|
6249
|
If goodness is constituted by God's will, it is a tautology to say God's will is good [Hutcheson]
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|
Full Idea:
To call the laws of the supreme Deity good or holy or just, if these be constituted by laws, or the will of a superior, must be an insignificant tautology, amounting to no more than 'God wills what he wills' or 'His will is conformable to his will'.
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|
From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §VII.V)
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|
A reaction:
This argues not only against God as the source of morality, but also against any rules, such as those of the Categorical Imperative. Why should I follow the Categorical Imperative? What has value must dictate the rules. Is obedience the highest value?
|