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Single Idea 9937
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
]
Full Idea
I do not believe mathematics either has or needs 'foundations'.
Gist of Idea
I do not believe mathematics either has or needs 'foundations'
Source
Hilary Putnam (Mathematics without Foundations [1967])
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.295
A Reaction
Agreed that mathematics can function well without foundations (given that the enterprise got started with no thought for such things), the ontology of the subject still strikes me as a major question, though maybe not for mathematicians.
The
138 ideas
from Hilary Putnam
3460
|
Superactors and superspartans count against behaviourism
[Putnam, by Searle]
|
11904
|
Express natural kinds as a posteriori predicate connections, not as singular terms
[Putnam, by Mackie,P]
|
11908
|
Putnam bases essences on 'same kind', but same kinds may not share properties
[Mackie,P on Putnam]
|
17505
|
Using proper names properly doesn't involve necessary and sufficient conditions
[Putnam]
|
17506
|
I now think reference by the tests of experts is a special case of being causally connected
[Putnam]
|
17507
|
Natural kind stereotypes are 'strong' (obvious, like tiger) or 'weak' (obscure, like molybdenum)
[Putnam]
|
17508
|
Science aims at truth, not at 'simplicity'
[Putnam]
|
18890
|
Putnam smuggles essentialism about liquids into his proof that water must be H2O
[Salmon,N on Putnam]
|
11192
|
If causes are the essence of diseases, then disease is an example of a relational essence
[Putnam, by Williams,NE]
|
3893
|
Often reference determines sense, and not (as Frege thought) vice versa
[Putnam, by Scruton]
|
7705
|
The Twin Earth theory suggests that intentionality is independent of qualia
[Jacquette on Putnam]
|
4099
|
If Twins talking about 'water' and 'XYZ' have different thoughts but identical heads, then thoughts aren't in the head
[Putnam, by Crane]
|
12026
|
We say ice and steam are different forms of water, but not that they are different forms of H2O
[Forbes,G on Putnam]
|
3208
|
Does 'water' mean a particular substance that was 'dubbed'?
[Putnam, by Rey]
|
11190
|
Archimedes meant by 'gold' the hidden structure or essence of the stuff
[Putnam]
|
11191
|
The hidden structure of a natural kind determines membership in all possible worlds
[Putnam]
|
9937
|
I do not believe mathematics either has or needs 'foundations'
[Putnam]
|
9941
|
Science requires more than consistency of mathematics
[Putnam]
|
9943
|
You can't deny a hypothesis a truth-value simply because we may never know it!
[Putnam]
|
9940
|
Maybe mathematics is empirical in that we could try to change it
[Putnam]
|
9939
|
It is conceivable that the axioms of arithmetic or propositional logic might be changed
[Putnam]
|
9944
|
We understand some statements about all sets
[Putnam]
|
9168
|
I can't distinguish elm trees, but I mean by 'elm' the same set of trees as everybody else
[Putnam]
|
5817
|
Language is more like a cooperative steamship than an individual hammer
[Putnam]
|
5818
|
If water is H2O in the actual world, there is no possible world where it isn't H2O
[Putnam]
|
5819
|
Conceivability is no proof of possibility
[Putnam]
|
9169
|
A statement can be metaphysically necessary and epistemologically contingent
[Putnam]
|
5820
|
'Water' has an unnoticed indexical component, referring to stuff around here
[Putnam]
|
9170
|
We need to recognise the contribution of society and of the world in determining reference
[Putnam]
|
5495
|
Instances of pain are physical tokens, but the nature of pain is more abstract
[Putnam, by Lycan]
|
6267
|
A culture needs to admit that knowledge is more extensive than just 'science'
[Putnam]
|
6266
|
We need the correspondence theory of truth to understand language and science
[Putnam]
|
6264
|
In Tarski's definition, you understand 'true' if you accept the notions of the object language
[Putnam]
|
6265
|
Tarski has given a correct account of the formal logic of 'true', but there is more to the concept
[Putnam]
|
6268
|
The claim that scientific terms are incommensurable can be blocked if scientific terms are not descriptions
[Putnam]
|
6269
|
Only Tarski has found a way to define 'true'
[Putnam]
|
6270
|
The correct translation is the one that explains the speaker's behaviour
[Putnam]
|
6271
|
How reference is specified is not what reference is
[Putnam]
|
6273
|
Knowledge depends on believing others, which must be innate, as inferences are not strong enough
[Putnam]
|
6274
|
Empathy may not give knowledge, but it can give plausibility or right opinion
[Putnam]
|
6272
|
'True' and 'refers' cannot be made scientically precise, but are fundamental to science
[Putnam]
|
17084
|
You can't decide which explanations are good if you don't attend to the interest-relative aspects
[Putnam]
|
6283
|
Language maps the world in many ways (because it maps onto other languages in many ways)
[Putnam]
|
6282
|
Theory of meaning presupposes theory of understanding and reference
[Putnam]
|
6281
|
Truth conditions can't explain understanding a sentence, because that in turn needs explanation
[Putnam]
|
6284
|
If a tautology is immune from revision, why would that make it true?
[Putnam]
|
6280
|
Realism is a theory, which explains the convergence of science and the success of language
[Putnam]
|
6276
|
'The rug is green' might be warrantedly assertible even though the rug is not green
[Putnam]
|
6277
|
Correspondence between concepts and unconceptualised reality is impossible
[Putnam]
|
6278
|
We should reject the view that truth is prior to meaning
[Putnam]
|
6275
|
You can't say 'most speaker's beliefs are true'; in some areas this is not so, and you can't count beliefs
[Putnam]
|
6279
|
A private language could work with reference and beliefs, and wouldn't need meaning
[Putnam]
|
13655
|
The Löwenheim-Skolem theorems show that whether all sets are constructible is indeterminate
[Putnam, by Shapiro]
|
9913
|
The Löwenheim-Skolem Theorem is close to an antinomy in philosophy of language
[Putnam]
|
9914
|
It is unfashionable, but most mathematical intuitions come from nature
[Putnam]
|
9915
|
V = L just says all sets are constructible
[Putnam]
|
6376
|
Neuroscience does not support multiple realisability, and tends to support identity
[Polger on Putnam]
|
2330
|
If humans and molluscs both feel pain, it can't be a single biological state
[Putnam, by Kim]
|
2587
|
Temperature is mean molecular kinetic energy, but they are two different concepts
[Putnam]
|
2588
|
Is pain a functional state of a complete organism?
[Putnam]
|
2589
|
Functionalism is compatible with dualism, as pure mind could perform the functions
[Putnam]
|
2590
|
Dispositions need mental terms to define them
[Putnam]
|
2591
|
Total paralysis would mean that there were mental states but no behaviour at all
[Putnam]
|
2592
|
Functional states correlate with AND explain pain behaviour
[Putnam]
|
17645
|
An alien might think oxygen was the main cause of a forest fire
[Putnam]
|
17644
|
Metaphysical realism is committed to there being one ultimate true theory
[Putnam]
|
17648
|
It is an illusion to think there could be one good scientific theory of reality
[Putnam]
|
17643
|
Shape is essential relative to 'statue', but not essential relative to 'clay'
[Putnam]
|
17642
|
The old view that sense data are independent of mind is quite dotty
[Putnam]
|
18199
|
Indispensability strongly supports predicative sets, and somewhat supports impredicative sets
[Putnam]
|
18200
|
Very large sets should be studied in an 'if-then' spirit
[Putnam]
|
8857
|
We must quantify over numbers for science; but that commits us to their existence
[Putnam]
|
3663
|
How can you contemplate Platonic entities without causal transactions with them?
[Putnam]
|
18949
|
The universal syllogism is now expressed as the transitivity of subclasses
[Putnam]
|
18951
|
For scientific purposes there is a precise concept of 'true-in-L', using set theory
[Putnam]
|
18950
|
Physics is full of non-physical entities, such as space-vectors
[Putnam]
|
18952
|
'⊃' ('if...then') is used with the definition 'Px ⊃ Qx' is short for '¬(Px & ¬Qx)'
[Putnam]
|
18954
|
Before the late 19th century logic was trivialised by not dealing with relations
[Putnam]
|
18956
|
Asserting first-order validity implicitly involves second-order reference to classes
[Putnam]
|
18953
|
Modern notation frees us from Aristotle's restriction of only using two class-names in premises
[Putnam]
|
18955
|
Having a valid form doesn't ensure truth, as it may be meaningless
[Putnam]
|
18957
|
Nominalism only makes sense if it is materialist
[Putnam]
|
18958
|
In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type
[Putnam]
|
18959
|
Sets larger than the continuum should be studied in an 'if-then' spirit
[Putnam]
|
18960
|
Most predictions are uninteresting, and are only sought in order to confirm a theory
[Putnam]
|
18961
|
We can identify functions with certain sets - or identify sets with certain functions
[Putnam]
|
18962
|
Unfashionably, I think logic has an empirical foundation
[Putnam]
|
4716
|
The correspondence theory is wrong, because there is no one correspondence between reality and fact
[Putnam, by O'Grady]
|
4714
|
Putnam's epistemic notion of truth replaces the realism of correspondence with ontological relativism
[Putnam, by O'Grady]
|
14214
|
If we try to cure the abundance of theories with causal links, this is 'just more theory'
[Putnam, by Lewis]
|
4718
|
If necessity is always relative to a description in a language, then there is only 'de dicto' necessity
[Putnam, by O'Grady]
|
7624
|
The word 'inconsiderate' nicely shows the blurring of facts and values
[Putnam]
|
7611
|
Rationality is one part of our conception of human flourishing
[Putnam]
|
7610
|
A fact is simply what it is rational to accept
[Putnam]
|
7612
|
Reference is social not individual, because we defer to experts when referring to elm trees
[Putnam]
|
7613
|
Concepts are (at least in part) abilities and not occurrences
[Putnam]
|
14200
|
'Water' on Twin Earth doesn't refer to water, but no mental difference can account for this
[Putnam]
|
14204
|
Naïve operationalism would have meanings change every time the tests change
[Putnam]
|
14206
|
There are infinitely many interpretations of a sentence which can all seem to be 'correct'
[Putnam]
|
14202
|
Neither individual nor community mental states fix reference
[Putnam]
|
14201
|
Maybe the total mental state of a language community fixes the reference of a term
[Putnam]
|
14205
|
The sentence 'A cat is on a mat' remains always true when 'cat' means cherry and 'mat' means tree
[Putnam]
|
14203
|
Intension is not meaning, as 'cube' and 'square-faced polyhedron' are intensionally the same
[Putnam]
|
14207
|
If cats equal cherries, model theory allows reinterpretation of the whole language preserving truth
[Putnam]
|
7617
|
Before Kant, all philosophers had a correspondence theory of truth
[Putnam]
|
7616
|
Truth is an idealisation of rational acceptability
[Putnam]
|
7618
|
Very nominalistic philosophers deny properties, though scientists accept them
[Putnam]
|
7620
|
Some kind of objective 'rightness' is a presupposition of thought itself
[Putnam]
|
7623
|
For ancient Greeks being wise was an ethical value
[Putnam]
|
8828
|
Truth is rational acceptability
[Putnam]
|
2334
|
Meaning holism tried to show that you can't get fixed meanings built out of observation terms
[Putnam]
|
2335
|
Understanding a sentence involves background knowledge and can't be done in isolation
[Putnam]
|
2336
|
Holism seems to make fixed definition more or less impossible
[Putnam]
|
2338
|
Reference (say to 'elms') is a social phenomenon which we can leave to experts
[Putnam]
|
2339
|
Aristotle implies that we have the complete concepts of a language in our heads, but we don't
[Putnam]
|
2343
|
Reference may be different while mental representation is the same
[Putnam]
|
2340
|
We should separate how the reference of 'gold' is fixed from its conceptual content
[Putnam]
|
2341
|
Like names, natural kind terms have their meaning fixed by extension and reference
[Putnam]
|
2342
|
"Water" is a natural kind term, but "H2O" is a description
[Putnam]
|
2344
|
If we are going to eliminate folk psychology, we must also eliminate folk logic
[Putnam]
|
2345
|
Semantic notions do not occur in Tarski's definitions, but assessing their correctness involves translation
[Putnam]
|
2346
|
Meaning and translation (which are needed to define truth) both presuppose the notion of reference
[Putnam]
|
2347
|
Asserting the truth of an indexical statement is not the same as uttering the statement
[Putnam]
|
2348
|
Is there just one computational state for each specific belief?
[Putnam]
|
2349
|
Realists believe truth is correspondence, independent of humans, is bivalent, and is unique
[Putnam]
|
2351
|
Aristotle says an object (e.g. a lamp) has identity if its parts stay together when it is moved
[Putnam]
|
2352
|
The job of the philosopher is to distinguish facts about the world from conventions
[Putnam]
|
2354
|
"Meaning is use" is not a definition of meaning
[Putnam]
|
2331
|
Functionalism says robots and people are the same at one level of abstraction
[Putnam]
|
2332
|
Functionalism can't explain reference and truth, which are needed for logic
[Putnam]
|
2074
|
Can we give a scientific, computational account of folk psychology?
[Putnam]
|
2071
|
If concepts have external meaning, computational states won't explain psychology
[Putnam]
|
10066
|
Putnam coined the term 'if-thenism'
[Putnam, by Musgrave]
|
2605
|
If everything uses mentalese, ALL concepts must be innate!
[Putnam]
|
2606
|
No machine language can express generalisations
[Putnam]
|
10269
|
Mathematics eliminates possibility, as being simultaneous actuality in sets
[Putnam]
|
22181
|
Putnam says anti-realism is a bad explanation of accurate predictions
[Putnam, by Okasha]
|
6782
|
Realism is the only philosophy of science that doesn't make the success of science a miracle
[Putnam]
|