Abstract
Mathematics, like the sciences, proceeds by a process of abstraction, so that mathematical claims like scientific claims are neither true nor false, but only true or false in an application of the theory to which they belong. A proof in mathematics is meant to show that a claim follows from the assumptions of a particular mathematical theory.
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Notes
- 1.
See Appendix 1 for a further discussion of this.
- 2.
We recognize this in English with the distinction between count and non-count (mass) terms. See (Epstein, 1994) for a discussion of the concept of ‘thing’ in our reasoning.
- 3.
Benjamin Lee Whorf says: Our tongue makes no distinction between numbers counted on discrete entities and numbers that are simply ‘counting itself.’ Habitual thought then assumes that in the latter the numbers are just as much counted on ‘something’ as in the former. This is objectification. (Whorf, 1941, 140)
In a similar vein, Aristotle says:
Whether if soul did not exist time would exist or not, is a question that may fairly be asked; for if there cannot be some one to count there cannot be anything that can be counted, so that evidently there cannot be number; for number is either what has been, or what can be, counted. (Aristotle, 1930, Book IV, 223a)
- 4.
Reuben Hersh (2006) discusses this issue from a similar perspective.
- 5.
See the presentation by Carnielli and me in (Epstein and Carnielli, 1989)
- 6.
Compare the presentation of Euclidean geometry in (Epstein, 2006). Albert Einstein says:
Geometry sets out from certain conceptions such as ‘plane,’ ‘point,’ and ‘straight line,’ with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as ‘true.’ Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e., they are proven. A proposition is then correct (‘true’) when it has been derived in the recognised manner from the axioms. The question of the ‘truth’ of the individual geometrical propositions is thus reduced to one of the ‘truth’ of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning. We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called ‘straight lines,’ to each of which is ascribed the property of being uniquely determined by two points situated on it. The concept ‘true’ does not tally with the assertions of pure geometry, because by the word ‘true’ we are eventually in the habit of designating always the correspondence with a ‘real’ object; geometry, however, is not concerned with the relation of ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.
It is not difficult to understand why, in spite of this, we feel constrained to call the propositions of geometry ‘true.’ Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of these ideas. (Einstein, 1915, 1–2).
- 7.
Though I criticize Hersh here, there is much good in his book, and reading it stimulated me to write this paper.
- 8.
See the axiomatization due to Lesław Szczerba of Euclidean plane geometry in (Epstein, 2006). Szczerba assures me that no one has a simpler independent axiomatization.
- 9.
See (Davies, 2005) for fuller discussions of these and more examples of methods of proof that are currently in debate in the mathematical community.
- 10.
- 11.
The picture comes from (Nelsen, 1993, 69). Martin Gardner, in (Gardner, 1973) from which Nelsen takes this example, says:
The first n consecutive positive integers can be depicted by dots in triangular formation. Two such triangles fit together to form a rectangular array containing n(n + 1) dots. Because each triangle is half of the rectangle, we see at once that the formula for the number of dots in each triangle is \(n(n + 1)/2\). This simple proof goes back to the ancient Greeks (Gardner, 1973, 114).
- 12.
A colleague said that the picture is convincing for n = 6 but not for any larger numbers. But then why not say it is good only for circles colored and laid out in this manner? See (Sherry, 2009) for a fuller discussion of this point.
- 13.
We can easily see that the method does not depend on these specific triangles and rectangles. Indeed, in ancient times often just one example, such as using ‘5’ and ‘8’ here (which is the ratio as close as the printer or computer monitor can manage), was given and the reader was expected to see that the method of proof or calculation was quite general. See (Bashmakova and Smirnova, 2000).
The issue of when we are justified in understanding specific parts of a diagram as variable is examined in (Kulpa, 2009). The counterpart in reasoning with formulas is given a precise answer in classical predicate logic by the theorem on constants: if a name appears in a theorem but does not appear in the axioms, then it can be replaced by a variable that is quantified universally (Epstein, 2006, Lemma 10.b). Kulpa’s paper also surveys recent work on formalizing reasoning with diagrams.
- 14.
See (Breger, 2000) for a discussion of this point and more on the nature of abstraction in mathematics.
- 15.
As I show with examples in (Epstein, 2006), first-order logic is rarely appropriate because much mathematics requires second-order assumptions. Second-order logic and set-theory give no unique standard because there are many systems of those that differ too much. Further, all those systems are based on a metaphysics that denies a mathematics of process as distinct from things (see Epstein, 2010). Moreover, it is quite common to establish a theorem in a formalized theory by semantic means rather than with a syntactic proof, as can be seen in (Epstein, 2006).
- 16.
The translation here comes from (Dauben, 1979, 128–129).
- 17.
Frege (1980, 39–40). Frege strongly disputes Hilbert’s view (op. cit., 43–47).
- 18.
This issue in historical context is discussed in (Detlefsen, 2005).
- 19.
This is how I developed the general framework for semantics in (Epstein, 1990). Rather than viewing propositional logics as about abstract things called ‘propositions,’ I saw them as ways to reason using ordinary language and abstractions of that. Rather than looking for the ‘right’ logic that captures exactly the properties of such abstract things, I saw that what we pay attention to in our abstracting, what aspects of ordinary language claims we deem important, determines the appropriateness of the logic we choose. As we vary the aspect we deem important, we vary the logic. That variation, I saw, can be described by devising an abstraction of our propositional logics—an abstraction of our abstractions. The general structures that arise are then worth investigating, not only for their own interest but for the relations among logics they illuminate and for the assumptions about how to reason well they uncover. I did not deny the abstract, for how could I show that there are no such things as abstract propositions? Rather, I focused on the process of abstracting, and that gave rise to new mathematics that is grounded in experience. See (Epstein, 1988) for a summary.
This is also what I do in (Epstein, 2006). I present formal logic as an abstraction of reasoning. By considering propositions as actual utterances or inscriptions rather than making the abstraction to treat different utterances as the same thing, I give a formal logic that resolves the liar paradox. By considering how we assign truth-values to atomic propositions rather than assuming that they come with truth-values, I develop a simple modification of classical mathematical logic that deals with names that do not refer.
- 20.
- 21.
For a comprehensive discussion of current views of mathematics see the excellent survey in the introduction to (Ferreirós and Gray, 2006).
- 22.
In (Putnam, 1967) he says that his view of mathematics as modal logic is equivalent to taking mathematics as based on set theory. In (Putnam, 1975, 72) he says: The main question we must speak to is simply, what is the point? Given that one can either take modal notions as primitive and regard talk of mathematical existence as derived, or the other way around, what is the advantage to taking the modal notions as the basic ones?
- 23.
John Skorupski (2005) discusses Mill’s views and suggests a program for empiricists that is similar to what I have done here.
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Acknowledgements
A previous version of this chapter was published as ‘On Mathematics’ in Richard L. Epstein, Logic as the Art of Reasoning Well, Advanced Reasoning Forum, 2008, 411–441. I am grateful to Fred Kroon, Charlie Silver, Carlo Cellucci, Jeremy Avigad, David Isles, Reuben Hersh, Ian Grant, Paul Livingston, and Andrew Aberdein for their comments on earlier versions.
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Epstein, R.L. (2013). Mathematics as the Art of Abstraction. In: Aberdein, A., Dove, I. (eds) The Argument of Mathematics. Logic, Epistemology, and the Unity of Science, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6534-4_14
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