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  1. Are there mathematical theories that do not use intuitive integers? [That is, do not use integers to write statements.]

  2. Can you propose a theory that describes natural integers, without using intuitive integers to state axioms, definitions, and propositions?

Sam Nead
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Dattier
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6 Answers6

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"Are there mathematical theories that do not use intuitive integers?"

Geometry? A lot of Euclid's Elements is done without numbers.

Gerald Edgar
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    How do you state "Two distinct points determine a line" without the number two? – Kapil Apr 18 '21 at 02:51
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    @Kapil $\forall x,y\exists!\ell(x\neq y\implies {x,y}\subseteq\ell)$? – Alec Rhea Apr 18 '21 at 04:28
  • Don't you need an algorithm to tell you to stop after two points? Mark point when sun sets, mark point when sun rises, connect morning point with evening point. :-) – PatrickT Apr 18 '21 at 08:23
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    @AlecRhea Nitpick: you need to put the condition $x\neq y$ before the $\exists!$ quantifier. :0) – Alex Kruckman Apr 18 '21 at 12:07
  • @AlecRhea One of the "missing" axioms of Euclidean geometry is that there are at least 3 non-collinear points and another states that each line contains at least two points. However, you have encapsulated the axiom I asked about! – Kapil Apr 18 '21 at 15:52
  • @Kapil: I would actually argue (related to the point in my answer) that axiomatic euclidean geometry requires you to already believe in natural numbers. If not, there is no guarantee that you can always draw a line between two drawn points and draw a point at intersection of two drawn lines. Because once there are at least 4 points in general position, you can draw infinitely many more if the concept of counting holds up. In other words, if you do not believe all naturals are constructible in some sense, then you cannot believe both of those construction axioms of euclidean geometry. – user21820 Apr 20 '21 at 12:49
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A "philosophy of math" tag would have been a good idea.

To answer your question, take a look at Hatry Field's "Science without Numbers". Also, Edward Nelson has developed a theory of "proto-integers", which is discussed in "Diffusion, Quantum Theory, and Radically Elementary Mathematics" (MN-47).

In my opinion, most attempts at nominalism (be it about abstract objects, infinite sets, or numbers) spend a lot of energy, and for what in the end? Moreover, the approach invariably comes across as a cheap trick (to me). This is my personal opinion, but I get the impression it is shared by many mathematicians.

I would quote Alonzo Church here, but the current climate does not allow for that anymore.

Sam Sanders
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    Hartry Field was the first name that came to my mind as well. As far as I can tell, his book largely reduces to a single claim: Numbers are abstract, but regions of space are concrete. Like most mathematicians, I can't make any sense of this claim. Along similar lines, Mary Leng (as I see it) claims that arithmetic is abstract, but logic is concrete. Say what? I don't get it. I discussed Leng's ideas on the Foundations of Mathematics mailing list about a year ago. – Timothy Chow Apr 17 '21 at 13:36
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    Why does the current climate not allow for quoting Church? – provocateur Apr 17 '21 at 13:37
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    Imagined misogyny for "The Ontological Status of Women and Abstract Entities"? Stretching it a bit IMHO – J.J. Green Apr 17 '21 at 14:26
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    I see no reason you can't quote Church here. – Tim Campion Apr 17 '21 at 15:07
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    As far as I know, Hartry Field's reduction result is considered pretty 'weak', as one generally needs numbers in the meta-theory to do anything real. This is also the problem with Nelson's proto-integers: you win an infinitesimal of "fewer abstract objects" at the cost of a huge overhead. Would Lang's claims (as you put them) not be easily refuted via experiments in psychology? Has this been done? – Sam Sanders Apr 17 '21 at 17:53
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    I have seen people be roundly abused for quoting Church, the talk mentioned by J.J. Green in particular, even during a discussion on ontological parsimony, i.e. "on topic". – Sam Sanders Apr 17 '21 at 19:32
  • @SamSanders I don't see how Leng's claims could be refuted by psychological experiments, but this comment section is perhaps not the right place for that discussion. – Timothy Chow Apr 17 '21 at 22:05
  • @Timothy Chow. You said "Mary Leng (as I see it) claims that arithmetic is abstract, but logic is concrete." The most basic experiment I can think of is just to ask many people questions like "which one of the following is more abstract: 2+2=4, $A\rightarrow B\equiv \neg A \vee B$, ...". Perhaps there is already a measure of "abstract vs concrete" in psychology, and one can also use that. Anyway, this is indeed beyond a simple comment. – Sam Sanders Apr 17 '21 at 22:12
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    @SamSanders Your experiment would explore a linguistic fact about the English words "concrete" and "abstract" as they are commonly used, but I was using the words in the technical sense of what nominalists consider real and unreal. – Timothy Chow Apr 17 '21 at 23:04
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    @Timothy Chow. I see. Thanks for the clarification. I must say that it has been a while since I dabbled in the philosophy of mathematics. – Sam Sanders Apr 18 '21 at 08:14
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Here are two attempts to answer what I view is the underlying question. The first attempt is taken from "Some Philosophical Prolegomena" (a section of the notes "The Axioms of Set Theory", by Tom Foster).

Many people come to set theory having been sold a story about its foundational significance; such people are often worried by apparent circularities such as...

Before we even reach set theory we have to have the language of first-order logic. Now the language of first-order logic is an inductively defined set and as such is the minimal [with respect to inclusion] set satisfying certain closure properties, and wasn’t it in order to clarify things like this (among others) that we needed set theory . . . ? And how can we talk about arities if we don’t already have arithmetic? And weren’t we supposed to get arithmetic from set theory?

...this doesn’t mean that set theory cannot serve as a foundation for Mathematics, but it does make the point that the whole foundation project is a bit more subtle than one might expect, and that the cirularities which launched this digression are not really pathologies, but a manifestation of the fact that life is complicated.

I have taken various liberties in selecting these quotations - you may want to consult the original text.

The second attempt is my own, more pessimistic, view. I think that "meaning" (if it exists at all) comes from a very lengthy bootstrapping process; thus any attempt at mathematical foundations is hopeless.

Sam Nead
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This question, like many others, is based on a fundamental conceptual error. Any formal system whatsoever that humans can devise and use must be one with unambiguous rules. Therefore its theorems (i.e. statements that it allows you to deduce) must necessarily be a computably decidable set. But even basic logic relies crucially on the assumption that any finite symbol strings can be concatenated. Otherwise you cannot even have a rule that allows you to deduce "A∨B" from "A"!

Because of this, the answer to your questions are trivially "no" and "no".

user21820
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There is a famous anecdot concerning Grothendieck and a certain integer. In the sixties, Alexander Grothendieck was developping a complex algebro-arithmetic framework in order to prove a set of conjectures due to Weil concerning varieties over finite fields. One motivation was to get a better grasp of integer solutions of polynomial equations using reduction mod p.

The lectures given by Grothendieck were very abstract and one of the auditor asked for an example. "Sure," said Grothendieck, "let's take some prime number p, for example 57". And so now 57 is known as the Grothendieck prime.

In modern algebraic geometry, an prime integer is a maximal ideal in the ring $\bf Z$. I don't remember what $\bf Z$ is exactly. Some algebraic geometer may come forward and correct me if I am wrong but I think that this is the terminal object in the category of scheme. Or maybe the antediluvian one. Another fashionable topic at the moment is the field with one element. The number 1 of course should not be understood in its intuitive meaning. All jokes taken aside, some parts of arithmetics nowadays are pretty sophisticated and do not rest on an intuitive notion of integers.

6 for example should be interpreted as a global section of the canonical sheaf of $spec({\bf Z})$, that is as a kind of function that associates to each point p in $spec({\bf Z})$ (a prime number actually) an element of ${\bf Z} / p{\bf Z}$ (the stalk of the sheaf at p) obtained by reducing 6 mod p. This allows to draw a parallel with the polynomial ring ${\bf C}[X]$, which happens to be a ring of functions over the complex line, and use idea coming from analytic geometry to study integers.

At that point, you should be confused enough to understand that integers are not what they used to be.

coudy
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  • You are right that $\mathrm{Spec}(\mathbb{Z})$ is the terminal object in the category of schemes. So $\mathbb{Z}$ is the initial object in $\mathbf{CRing}$ (since $\mathbf{CRing}$ is dual to the category of affine schemes). – xuq01 Nov 22 '21 at 00:11
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Topology and group theory do not use the integers.

However, I'd say that groups are - one of many - generalisations of the integers.

Also set theory and as one poster has already pointed out, Euclidean geometry.

Mozibur Ullah
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  • How do you say that $card (G) = 10^9$ without using intuitive integers? – Dattier Apr 18 '21 at 12:44
  • @Dattier: Obviously, mathematical disciplines are inter-relate. For example, group theory is used in geometry in the definition of Lie groups and principal bundles. However, we can restrict usage. It's possible to build up a useful set theory without using the notion of cardinality. For example, the de Morgan identities can be shown etc. – Mozibur Ullah Apr 18 '21 at 12:51
  • Well, can you define $10^9$ in arithmetic of Peano without use intuitive integers and knowing BAAC =BAC for BAAC and BAC a set of symbol concatenate – Dattier Apr 18 '21 at 12:58
  • @Dattier: Intuitive set theory, geometry and topology can be done without intuitive integers. Given that the OP was asking about intuitive integers I figured that simplicity is best. Peano models counting axiomatically and the OP is asking about mathematics other than counting. – Mozibur Ullah Apr 18 '21 at 14:09
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    I'm not sure how one talks about the winding number without integers, and I'm not sure how one does topology without the winding number. – Dan Fox Nov 15 '21 at 13:48
  • @Dan Fox: Well, I assume you don't talk about them. I'm not saying that doing without the integers is useful. Its not. But I am saying it is possible to do without them. Do you need the integers to express the separation axioms? How about the notion of continuity or an open set? These are all useful notions that don't use integers. – Mozibur Ullah Dec 12 '21 at 09:47