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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics

[maths as a necessity for empirical investigation]

17 ideas
If it can't be expressed mathematically, it can't occur in nature? [Heisenberg]
     Full Idea: The solution was to turn around the question How can one in the known mathematical scheme express a given experimental situation? and ask Is it true that only such situations can arise in nature as can be expressed in the mathematical formalism?
     From: Werner Heisenberg (Physics and Philosophy [1958], 02)
     A reaction: This has the authority of the great Heisenberg, and is the ultimate expression of 'mathematical physics', beyond anything Galileo or Newton ever conceived. I suppose Pythagoras would have thought that Heisenberg was obviously right.
Mathematics is part of science; transfinite mathematics I take as mostly uninterpreted [Quine]
     Full Idea: The mathematics wanted for use in empirical sciences is for me on a par with the rest of science. Transfinite ramifications are on the same footing as simplifications, but anything further is on a par rather with uninterpreted systems,
     From: Willard Quine (Review of Parsons (1983) [1984], p.788), quoted by Penelope Maddy - Naturalism in Mathematics II.2
     A reaction: The word 'uninterpreted' is the interesting one. Would mathematicians object if the philosophers graciously allowed them to continue with their transfinite work, as long as they signed something to say it was uninterpreted?
Nearly all of mathematics has to quantify over abstract objects [Quine]
     Full Idea: Mathematics, except for very trivial portions such as very elementary arithmetic, is irredeemably committed to quantification over abstract objects.
     From: Willard Quine (Word and Object [1960], §55)
     A reaction: Personally I would say that we are no more committed to such things than actors in 'The Tempest' are committed to the existence of Prospero and Caliban (which is quite a strong commitment, actually).
Science requires more than consistency of mathematics [Putnam]
     Full Idea: Science demands much more of a mathematical theory than that it should merely be consistent, as the example of the various alternative systems of geometry dramatizes.
     From: Hilary Putnam (Mathematics without Foundations [1967])
     A reaction: Well said. I don't agree with Putnam's Indispensability claims, but if an apparent system of numbers or lines has no application to the world then I don't consider it to be mathematics. It is a new game, like chess.
Indispensability strongly supports predicative sets, and somewhat supports impredicative sets [Putnam]
     Full Idea: We may say that indispensability is a pretty strong argument for the existence of at least predicative sets, and a pretty strong, but not as strong, argument for the existence of impredicative sets.
     From: Hilary Putnam (The Philosophy of Logic [1971], p.346), quoted by Penelope Maddy - Naturalism in Mathematics II.2
We must quantify over numbers for science; but that commits us to their existence [Putnam]
     Full Idea: Quantification over mathematical entities is indispensable for science..., therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question.
     From: Hilary Putnam (The Philosophy of Logic [1971], p.57), quoted by Stephen Yablo - Apriority and Existence
     A reaction: I'm not surprised that Hartry Field launched his Fictionalist view of mathematics in response to such a counterintuitive claim. I take it we use numbers to slice up reality the way we use latitude to slice up the globe. No commitment to lines!
It is spooky the way mathematics anticipates physics [Weinberg]
     Full Idea: It is positively spooky how the physicist finds the mathematician has been there before him or her.
     From: Steven Weinberg (Lecture on Applicability of Mathematics [1986], p.725), quoted by Stewart Shapiro - Thinking About Mathematics 2.3
     A reaction: This suggests that mathematics might be the study of possibilities or hypotheticals, like mental rehearsals for physics. See Hellman's modal structuralism. Maybe mathematicians are reading the mind of God, but I doubt that.
Actual measurement could never require the precision of the real numbers [Bostock]
     Full Idea: We all know that in practice no physical measurement can be 100 per cent accurate, and so it cannot require the existence of a genuinely irrational number, rather than some of the rational numbers close to it.
     From: David Bostock (Philosophy of Mathematics [2009], 9.A.3)
Physics requires the existence of properties, and also the abstract objects of arithmetic [Rey]
     Full Idea: Physics is committed to arithmetic, which seems committed to abstract objects such as numbers, and its causal explanations seem to appeal to properties, such as mass and charge.
     From: Georges Rey (Contemporary Philosophy of Mind [1997], 2.3)
The application of mathematics only needs its possibility, not its truth [Field,H, by Shapiro]
     Full Idea: Field argues that to account for the applicability of mathematics, we need to assume little more than the possibility of the mathematics, not its truth.
     From: report of Hartry Field (Science without Numbers [1980]) by Stewart Shapiro - Philosophy of Mathematics 7.2
     A reaction: Very persuasive. We can apply chess to real military situations, provided that chess isn't self-contradictory (or even naturally impossible?).
Hilbert explains geometry, by non-numerical facts about space [Field,H]
     Full Idea: Facts about geometric laws receive satisfying explanations, by the intrinsic facts about physical space, i.e. those laid down without reference to numbers in Hilbert's axioms.
     From: Hartry Field (Science without Numbers [1980], 3)
     A reaction: Hilbert's axioms mention points, betweenness, segment-congruence and angle-congruence (Field 25-26). Field cites arithmetic and geometry (as well as Newtonian mechanics) as not being dependent on number.
Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H]
     Full Idea: Field needs the notion of logical consequence in second-order logic, but (since this is not recursively axiomatizable) this is a semantical notion, which involves the idea of 'true in all models', a set-theoretic idea if there ever was one.
     From: comment on Hartry Field (Science without Numbers [1980], Ch.4) by James Robert Brown - Philosophy of Mathematics
     A reaction: Brown here summarises a group of critics. Field was arguing for modern nominalism, that actual numbers could (in principle) be written out of the story, as useful fictions. Popper's attempt to dump induction seemed to need induction.
We must treat numbers as existing in order to express ourselves about the arrangement of planets [Yablo]
     Full Idea: It is only by making as if to countenance numbers that one can give expression in English to a fact having nothing to do with numbers, a fact about stars and planets and how they are numerically proportioned.
     From: Stephen Yablo (Apriority and Existence [2000], §13)
     A reaction: To avoid the phrase 'numerically proportioned', he might have alluded to the 'pattern' of the stars and planets. I'm not sure which -ism this is, but it seems to me a good approach. The application is likely to precede the theory.
Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy]
     Full Idea: Crudely, the scientist posits only those entities without which she cannot account for observations, while the set theorist posits as many entities as she can, short of inconsistency.
     From: Penelope Maddy (Naturalism in Mathematics [1997], II.5)
Maybe applications of continuum mathematics are all idealisations [Maddy]
     Full Idea: It could turn out that all applications of continuum mathematics in natural sciences are actually instances of idealisation.
     From: Penelope Maddy (Naturalism in Mathematics [1997], II.6)
If a notion is ontologically basic, it should be needed in our best attempt at science [Schaffer,J]
     Full Idea: Science represents our best systematic understanding of the world, and if a certain notion proves unneeded in our best attempt at that, this provides strong evidence that what this notion concerns is not ontologically basic.
     From: Jonathan Schaffer (Causation and Laws of Nature [2008], 3.2)
     A reaction: But is the objective of science to find out what is 'ontologically basic'? If scientists can't get a purchase on a question, they have no interest in it. What are electrons made of?
Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend]
     Full Idea: Central to naturalism about mathematics are 'indispensability arguments', to the effect that some part of mathematics is indispensable to our best physical theory, and therefore we ought to take that part of mathematics to be true.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.1)
     A reaction: Quine and Putnam hold this view; Field challenges it. It has the odd consequence that the dispensable parts (if they can be identified!) do not need to be treated as true (even though they might follow logically from the dispensable parts!). Wrong!