Every foundational system for mathematics I have ever read about has been a set theory, from ETCS to ZFC to NF. Are there any proposals for a foundational system which is not, in any sense, a set theory? Is there any alternative foundation which is not a set-theory?
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1$\begingroup$ Question too vague (at present) to admit a proper answer. You haven't said what you mean by "not a set-theory"; nor does what you have written give any idea what an "answer" to your question would involve. $\endgroup$– Yemon ChoiCommented Dec 3, 2009 at 1:18
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1$\begingroup$ I've no idea what "some kind of discontinuity instead of object/set" is supposed to mean. Set theory was simply found to be an extremely useful foundation for much of modern mathematics (and that's to put it mildly). Why do you want another foundation? Why do you suggest that it most possess some different ontological assumption, whatever that means? Please note that this isn't a good place to have a discussion - if you're unable to frame a concrete mathematical question, that's fine, but you may have better success at other forums. $\endgroup$– Alon AmitCommented Dec 3, 2009 at 1:24
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2$\begingroup$ The short answer to "are there any alternative foundations exist" is no, not really. Set theory itself has a couple of different formalisms, but I don't know of anything else that was found to be useful as a foundation. $\endgroup$– Alon AmitCommented Dec 3, 2009 at 1:39
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3$\begingroup$ Bill Lawvere has suggested axiomatizing the category of categories as a foundation of mathematics, and there is no sense in which this could be thought of as a set theory. Colin McLarty is one person who has done some work on achieving such an axiomatization. $\endgroup$– Steven GubkinCommented Dec 3, 2009 at 11:17
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1$\begingroup$ This seems like a real question to me. It is of interest to me, but maybe it is Philosopher Steve, not Mathematician Steve who is interested. Obviously the question could be better posed. This would be my post (if I were psihodelia) "Every foundational system for mathematics I have ever read about has been a set theory, from ETCS to ZFC to NF. Are there any proposals for a foundational system which is not, in any sense, a set theory?" $\endgroup$– Steven GubkinCommented Dec 3, 2009 at 19:55
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10 Answers
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Bill Lawvere has suggested axiomatizing the category of categories as a foundation of mathematics, and there is no sense in which this could be thought of as a set theory. Colin McLarty is one person who has done some work on achieving such an axiomatization.
answered Dec 4, 2009 at 19:24
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2$\begingroup$ Somebody now seems to have asked/answered a question describing this in detail: mathoverflow.net/questions/8731/… $\endgroup$ Commented Dec 13, 2009 at 8:53
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If we adopt a historical attitude, then there is an extremely good answer, namely, Geometry. For approximately two thousand years, (Euclidean) geometry was taken to be the foundation of all mathematics. Numbers were regarded as lengths of line segments; quadratic equations were regarded as expressing the relationship of the areas of certain geometrical figures. All mathematics was, at bottom, geometric.
Of course, there were problems with this. For many ancient mathematicians, for example, it made as little sense to add the cube of a number to its square as it would to add a volume to an area. The equals sign was not introduced until 1557, and it is easy for contemporary mathematicians to lose sight of how differently the ancients thought about the mathematical objects they studied and wrote about. Newton, famously, was so great a mathematician that he was able to introduce the differential calculus in terms of the ancient geometrical reasoning, although we rarely conceive of his ideas that way today. Indeed, much of this kind of writing now appears alien to us.
answered Dec 5, 2009 at 20:26
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1$\begingroup$ Unfortunately Geometry, at least in the form axiomatised by Tarski (see en.wikipedia.org/wiki/Tarski%27s_axioms), lacks the expressive power needed to be a good foundation for other branches of mathematics. $\endgroup$ Commented Jan 18, 2010 at 21:00
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$\begingroup$ Has there been recent work that uses geometry as a foundation of mathematics? I'm very interested in pursue this further. $\endgroup$– user2529Commented May 16, 2010 at 9:34
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$\begingroup$ It may be too naive to formulate addition as concatenation of lengths/areas/volumes. Take something geometrically simple like translation for instance. It took till 1500 Descartes for this to be formulated properly in terms of coordinates (and vector spaces). It involves not just the idea of translation as addition or subtraction, but also the idea of triples of numbers to represent space. We must not rule out the possibility of geometry being able to formulate addition, not directly, not via layers and layers of symbolization, representation and ideas. $\endgroup$– user2529Commented May 16, 2010 at 9:42
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1$\begingroup$ @Colin: Using Desargues' Theorem one can define basic arithmetic operations (on the points of a line) geometrically. So one can use geometry as a foundation for some portions of arithmetic. As sigfpe notes, however, it seems that (synthetic) geometry alone is not powerful enough to be a foundation for all of mathematics unless you sneak in some set theory through the back door. $\endgroup$ Commented May 17, 2010 at 14:35
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1$\begingroup$ @DanPiponi, Euclidean geometry always included areas and figures not captured in Tarski's axiomatization, so Tarski's results don't provide a limit on its expressive power. Consider the generalized Pythagorean theorem in Euclid's Elements, Book VI, Prop 31: "In right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle", which is not a first-order sentence about lines and points. $\endgroup$– user44143Commented Nov 22, 2022 at 19:13
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Type theories form another class of foundations for mathematics, and are used in various places. For example, Martin-Löf type theory is a constructive foundation of mathematics, and a lot of constructive mathematics has been formulated in it.
Type theories are used in some proof assistants, like Coq, and they have nice connections with various programming languages in computer science - look up languages with dependent types.
I should also mention that type theories have a close relation with categories - "Introduction to higher order categorical logic" by Lambek and Scott connects various type theories and categories.
answered Dec 4, 2009 at 21:16
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4$\begingroup$ For a study comparing all three approaches see Steve Awodey's From sets, to types, to categories, to sets andrew.cmu.edu/user/awodey/preprints/stcsFinal.pdf $\endgroup$ Commented Dec 4, 2009 at 21:55
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3$\begingroup$ To me it is questionable whether type theories are really different from set theories, or whether they merely lie at opposite ends of a continuum. To be sure, it would be hard to recast ZFC as a type theory in the traditional sense, or System F as a set theory in the traditional sense, but there is considerable overlap in the middle. It's hard to distinguish ETCS, for instance, from the higher-order type theory that is the internal logic of a topos. $\endgroup$ Commented Dec 5, 2009 at 21:36
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$\begingroup$ @Mike do you still hold this view, that type theories may not be that different to set theories? $\endgroup$– David Roberts ♦Commented Dec 11, 2022 at 2:24
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$\begingroup$ Haha! No, not really. Although it depends a bit on what the OP meant by "in any sense". At a high philosophical level, one could argue that set-level type theory is a "set theory" in some sense, and I think that's what I had in mind when I wrote this back in the pre-HoTT days. This question could use an answer about HoTT. $\endgroup$ Commented Dec 11, 2022 at 3:31
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Arithmetic can be used as a foundation for a surprising amount of mathematics. The book Subsystems of Second-Order Arithmetic by Steve Simpson demonstrates that a huge fraction of mathematics can be formalized arithmetically. In fact, first-order Peano arithmetic suffices for most "ordinary" mathematics.
I should point out, however, that even when using arithmetic axioms as one's ultimate foundation, people in practice formalize everything in (finitary) set theory first, and then show how to encode finite sets as integers. Set theory is just so darn convenient as a unifying language that it's hard to get away from it entirely. However, as long as you're really only dealing with finite or countable sets, almost anything you want to state and prove can in principle be done with integers, so in this sense arithmetic can be used as a foundation for most of mathematics.
Areas of mathematics that are "intrinsically uncountable" cannot be captured by any of the systems in Simpson's book, but there are fewer of these areas than you might think.
answered May 16, 2010 at 3:52
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I believe the lambda calculus was originally intended as a foundation for mathematics. More recently it seems that both category theory and type theory seem to be gaining support. Although, I think type theory (a la Martil-Löf) could be viewed as another variation on the set-theoretic theme.
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This was going to be a comment to Joel David Hamkins's answer on geometry, but it didn't fit.
+1 This is one of the most clear-minded things I have read on MO. It does not make a mockery of Foundations and still says something non-obvious.
I'm very skeptical of all this business with category theory being a foundation for mathematics. First, whenever anyone talks about it, it always seems to be somebody else's work. It's become something of a meme that "Bill Lawvere has proposals to provide foundations for math through category theory", but we don't ever see details provided.
Second, are we really to understand that we are going to add small integers with arrows and diagrams? Draw circles and lines, and say that the latter meet at most once? I think people work in trans-Euclidean hyperschemes of infinite type so much, they forget that math includes these things.
As Wittgenstein remarks in the Investigations, just because you can express A in terms of B, it does not mean that B actually underlies A.
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8$\begingroup$ "Second, are we really to understand that we are going to add small integers with arrows and diagrams?" Think of the analogue for set theory: are we really to understand that a real number or a triangle is a set of sets of sets of sets ... of sets? $\endgroup$ Commented Mar 19, 2011 at 1:35
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6$\begingroup$ Regarding "all this business with category theory being a foundation", I'm not sure what your true grounds for skepticism are -- have you ever tried investigating this? If you asked a question on MO (phrased in suitably non-subjective, non-argumentative language, of course), you might get a thoughtful reply. One recent development pertaining to foundations and which connects homotopy theory, intensional type theory, and higher category theory, has been discussed recently at the n-Category Cafe. See especially the posts Homotopy Type Theory I and II. $\endgroup$ Commented Mar 22, 2011 at 22:19
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Von Neumann wrote down a foundation where the basic objects are functions, not sets. But it was soon re-cast into an equivalent system with sets (and classes).
answered Dec 4, 2009 at 23:51
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2$\begingroup$ Hey! I know you! Your my Analysis recitation instructor! $\endgroup$ Commented Dec 5, 2009 at 0:24
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$\begingroup$ @Gerald Lawvere and Rosebaugh do something very similar in "Sets for Mathematics",using functions as the elements of thier foundations.But they don't eliminate set theory either-they use the functions to build sets rather then the converse. Sets have the enormously important advantage of complete determinacy of thier composition and it's not entirely clear that a foundation for mathematics devoid of them would preserve this property. I think this would be very problematic for most classical mathematics. $\endgroup$ Commented Jun 9, 2010 at 16:41
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This may not satisfy the request for something that is "not, in any sense, a set theory" but Oliver Deiser has worked out two versions of foundations, one based on lists and one on multisets. This is in his book "Orte, Listen, Aggregate" (and his Habilitationsschrift with the same title).
answered Mar 22, 2011 at 21:06
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$\begingroup$ @Andreas Blass Thank You. Especially interested in lists (for computers lists are the basic, sets come afterwards: sets are equivalence classes of lists, or categorically, one 'forgets" from LIST into SET) I went to his web page, but could not find a downloadable summary of his foundational research. There seem to be a published book, which I may buy if nothing else is available for free. Any pointers? PS Werke auf deutsch wurden auch gut sein) $\endgroup$ Commented Jun 2, 2011 at 22:56
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$\begingroup$ @Mirco Mannucci: I've just e-mailed a copy of Oliver Deiser's Habilitationsschrift to you (or to someone with the same name as yours --- I got your e-mail address from Google because it's not in your MO profile). You might also be interested in a paper "Why Sets" that Yuri Gurevich and I wrote a few years ago. It's available at math.lsa.umich.edu/~ablass/set.pdf (and also at Yuri's web site at Microsoft). $\endgroup$ Commented Jun 3, 2011 at 0:04
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$\begingroup$ Correction: The right source for "Why Sets" is Yuri Gurevich's page, research.microsoft.com/en-us/um/people/gurevich/Opera/172.pdf because the version on my web page is an older one (written before Deiser's work). (One of these days, I'll update my web page.) $\endgroup$ Commented Jun 3, 2011 at 0:11
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To follow up further on Joel David Hamkins's answer on geometry, Frege’s last work (two despairing decades after Russell’s Paradox demolished his Grundgesetze der Arithmetik) was a brief unpublished paper entitled “Neuer Versuch der Grundlegun der Arithmetik,” based on geometry with “the final goal, the general complex numbers.” (As in the Grundgesetze, he emphasizes that real numbers are ratios of quantities, not quantities themselves.)
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$\begingroup$ Can you expand in what the general complex numbers are, how real numbers are ratios, and what they’re ratios of? $\endgroup$ Commented Dec 8, 2022 at 18:13
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I know from the logical end there is plural quantification developed and expounded to some extent by Boolos, Lewis and others that sidesteps the whole issue of set and gives first order logic the ability to talk about set-like objects without resorting to set theory.