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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?

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    $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – Tim Campion Apr 18 at 14:00
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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.

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    $\begingroup$ How do you state "Two distinct points determine a line" without the number two? $\endgroup$ – Kapil Apr 18 at 2:51
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    $\begingroup$ @Kapil $\forall x,y\exists!\ell(x\neq y\implies \{x,y\}\subseteq\ell)$? $\endgroup$ – Alec Rhea Apr 18 at 4:28
  • $\begingroup$ 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. :-) $\endgroup$ – PatrickT Apr 18 at 8:23
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    $\begingroup$ @AlecRhea Nitpick: you need to put the condition $x\neq y$ before the $\exists!$ quantifier. :0) $\endgroup$ – Alex Kruckman Apr 18 at 12:07
  • $\begingroup$ @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! $\endgroup$ – Kapil Apr 18 at 15:52
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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.

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    $\begingroup$ 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. $\endgroup$ – Timothy Chow Apr 17 at 13:36
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    $\begingroup$ Why does the current climate not allow for quoting Church? $\endgroup$ – provocateur Apr 17 at 13:37
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    $\begingroup$ Imagined misogyny for "The Ontological Status of Women and Abstract Entities"? Stretching it a bit IMHO $\endgroup$ – J.J. Green Apr 17 at 14:26
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    $\begingroup$ I see no reason you can't quote Church here. $\endgroup$ – Tim Campion Apr 17 at 15:07
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    $\begingroup$ 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". $\endgroup$ – Sam Sanders Apr 17 at 19:32
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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.

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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".

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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.

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  • $\begingroup$ How do you say that $card (G) = 10^9$ without using intuitive integers? $\endgroup$ – Dattier Apr 18 at 12:44
  • $\begingroup$ @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. $\endgroup$ – Mozibur Ullah Apr 18 at 12:51
  • $\begingroup$ 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 $\endgroup$ – Dattier Apr 18 at 12:58
  • $\begingroup$ @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. $\endgroup$ – Mozibur Ullah Apr 18 at 14:09

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