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I understand numbers to be defined as objects defined to have certain convenient properties in relation to certain operations. It is very surprising that the exact same group objects should be applicable to the modelling of such seemingly disparate concepts as length, area, degree, cardinality, time, wavefunction, etc. I also find it surprising that the qualities of numbers, which appear quite contrived, should correspond to anything real in the first place. How can I make the relationship between these concepts and the concept of number more intuitive and expected?

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    I wouldn't think of numbers as objects, but 'concept' is an appropriate word. Concept basically means: something we made up. So, can something we made up apply to something else we made up? Yes
    – Scott Rowe
    Sep 30, 2023 at 19:01
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    It's not intuitive nor expected at all, that the properties of natural numbers model anything in the real world at all. In many way's the model seems absurd. That Geometric Length ( with it's one dimensional nature) can be used to say anything about length in the real world is a mystery. I think, the easiest case to make is that we built numbers using our understanding of the world, which is why natural numbers share similarities with it. Sep 30, 2023 at 20:15
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    It's because if the fundamental property of continuous symmetry under transformation in our intersubjective experiences: 'The Unreasonable Ineffectiveness of Mathematics in most sciences' philosophy.stackexchange.com/questions/92058/…
    – CriglCragl
    Sep 30, 2023 at 20:36
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    Numbers are not "defined" by those properties even today, they are formally undefinable and sometimes are axiomatized to have them. But, historically, those properties and operations were selected exactly as a minimal commonality in counting/measuring all those disparate concepts. The same broad applicability is displayed by other abstract objects (sets, etc.) that philosophers call "representational aids", see Yablo, Go Figure:"It is no big surprise if things with representing as their reason for “being” show a consistent aptitude for the task."
    – Conifold
    Oct 1, 2023 at 4:08
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    All your examples are nothing but the category of quantities, per Kant your question expressed by your last sentence is the answer to itself since number is the very intuition of the category of quantities. Hegel perhaps would replace your choice of word 'intuition' with his subjective 'notion' of Hegelian dialectic logic. Since either intuition or notion are premises of Kantian transcendental ideal intuitionism, it cannot be made any more intuitive...From math POV set theory is popular foundation, mere real number closed field(s) theory is complete but cannot capture all concepts of reals... Oct 1, 2023 at 7:09

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You should not be surprised. Numbers were conceived to model concepts such as length, area, angles and so on. Any qualities that vary by extent can be compared in an analogous way. For example, the difference between one egg and two eggs is comparable with the difference between one apple and two apples, or one grain of sand and two grains, or a stick of one length and a stick of twice the length, so why should you be surprised that the idea of 'two' arises in connection with all of them?

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Number is a formal abstraction created specifically to do what they do. What I mean by formal abstraction is sometimes illustrated by parallel lines. What parallel lines have in common is that they all have the same direction. What do we mean by direction? Well, one way to account for direction is as a formal abstraction of the concept of being parallel. We will define direction as an operator on lines, $ such that if A is a line, then $A represents the direction of A.

But wait, that doesn't really define anything because lines have different properties; which property of A does $A pick out? Well, it is that property such that $A=$B if and only if A is parallel to B. We have just defined direction based on parallelism. When defined this way, direction is said to be an abstraction. I added "formal" to distinguish it from less formal notions of what an abstraction is. A formal abstraction is specifically an abstraction drawn from an equivalence relation such as "being parallel" in the way I just showed.

Here's another example. Let us say that two sets are equinumerous just in case there is a one-to-one correspondence between the sets. What I've just done is define what it means for two sets to have the same number of elements without referring to number. Equinumerousness is an equivalence relation, so we can create a formal abstraction #S, where S is a set and define #S=#T if and only if S and T are equinumerous sets. I've just defined natural numbers as a formal abstraction of equinumerousness. We can define as a+b=c if and only if a=#S, b=#T, and c=#U where S and T are disjoint sets and U is the union of S and T.

Everything I've said so far is from Frege, modified by later authors to avoid an inconsistency in Frege's original logic. The following is my own gloss:

We can define real numbers as a formal abstraction of what I call a cleavable type. Cleaving is a relationship between objects, objects that are continuous and can be broken up into continuous parts. For example, a line can be cleaved into smaller line segments, a mass can be cleaved into smaller masses. Cleaving is a formal relationship, not a physical process. There's a lot more to defining what cleaving is, but that's the general intuition. Now, define an equivalence relation, equimagnitude such that it represents, in some sense, the same amount of a cleavable type. Two lines are equivalent in length, two masses are equivalent in mass. Define an abstraction %M based on equimagnitude such that %M=%L if and only if M and L are equimagnitude. You can (I claim) develop real arithmetic from this abstraction based on a cleaving relation.

And that's essentially what real numbers are (or started out as). They represent an abstraction of anything that is continuous but can be divided up into continuous parts, and such that there is an equimagnitude relation on the things. So naturally, they will work when applied to anything that fits that description.

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  • +1 For the OP, Gudeman's neologism 'cleaving' might best be understood as a dividing a whole into parts in the mereological sense. It would be advantageous to understand the basics of mereology and how such an ontological framework for systems undergirds mathematical thinking as a whole.
    – J D
    Oct 1, 2023 at 16:38
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    It's different from a mereological fusion operator in that it doesn't allow for overlap. You need to require disjoint regions for real addition just like you need to require disjoint sets for natural addition. Oct 1, 2023 at 18:12
  • It wouldn't do to add the same 50 cents to make a dollar. But I think people have built in fraud detection for that. Even babies respond negatively to unfairness.
    – Scott Rowe
    Oct 2, 2023 at 1:28
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There are a host of different philosophical positions on what math is. The philosophy of math has positions such as Platonic realism, constructivism, structuralism, etc. Numbers are by some defined to be abstract objects (SEP), but not by all. To conceive of numbers as such is a metaphysical position. When discussing what exists and what it is like, this is ontological discourse. Famous ontologists like Plato, Aristotle, Meinong, Carnap, and Quine each have theories about what numbers are, and it takes a broad reading to get familiar with those positions, but a good start is Linnebo's Philosophy of Mathematics.

I'm going to answer from a position that is consistent with conceptualism or nominalism in mathematics (SEP) which presumes that numbers are not real things, and they are not objects in any sense implying they are real and independent of real things, but are language-embodied concepts which describe the experience of real things.

You say:

I also find it surprising that the qualities of numbers, which appear quite contrived, should correspond to anything real in the first place.

You have the cart before the horse. If survival means counting the number of predators to chase you, and we are evolved organisms, than our faculties are physically evolved to correspond to real things. Why would this be? One chooses a different strategy for evading one predator as opposed to two. Or in the case of comparing two sources of food, one prefers to secure more apples in the tree than fewer. Thus, numerical cognition has evolved in our brains to help us to deal with real things in the real world. In fact, long before people acquired number systems, they had evolved the capacity for subitization. From WP:

Subitizing is the rapid, accurate, and confident judgments of numbers performed for small numbers of items. The term was coined in 1949 by E. L. Kaufman et al., and is derived from the Latin adjective subitus (meaning "sudden") and captures a feeling of immediately knowing how many items lie within the visual scene, when the number of items present falls within the subitizing range. Sets larger than about four to five items cannot be subitized unless the items appear in a pattern with which the person is familiar (such as the six dots on one face of a die). Large, familiar sets might be counted one-by-one (or the person might calculate the number through a rapid calculation if they can mentally group the elements into a few small sets). A person could also estimate the number of a large set—a skill similar to, but different from, subitizing.

Thus numbers are not abstract objects in the sense they exist independent of us, but rather are properties we see in things. Two-ness is a property that can be applied to anything countable in distinction to things we see only as wholes. In English, this manifests itself grammatically as a dichotomy between mass and countable nouns.

You ask:

How can I make the relationship between these concepts and the concept of number more intuitive and expected?

Read conceptualization of numbers are that are consistent with a naturalized epistemology (SEP). One of the things that newcomers to philosophy might not be aware is that much early philosophical thinking, let's use Plato's view on mathematics for example, is pre-scientific. Of course, 2,500 years ago there was no modern science, since Bacon and Galileo had not yet come around to suggest ideas that lead to contemporary science. He can be excused for thinking numbers were somehow real. Today, cognitive science, for instance, backs an entirely different approach to conceptualizing numbers. It would be sort of irrational to develop a belief about numbers independent of what modern science tells us about the world. I would suggest that your difficulties actually stem from your scientific intuitions of the world colliding with very out-moded realist thinking about abstractions.

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  • Experiments with 6 month old babies showed that they could accurately subitize up to about 100, when shown two images with slightly different numbers of dots.
    – Scott Rowe
    Oct 2, 2023 at 1:23
  • @ScottRowe Experimenting on babies is always fun!
    – J D
    Oct 2, 2023 at 1:26
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I think that the more fundamental question is "why is our world so free" in the mathematical sense of free. For example, as others have pointed out, the natural numbers can be generated from zero and the successor operation. The integers mod 3 can also be generated from zero and successor but with the added rule that S(S(S(Z)))=Z. The construction of the natural numbers is free but the added rule for integers mod 3 makes that construction not free.

Similarly the rational numbers are the free field. Real numbers are just the rationals with limit points and the complex numbers are just the reals with roots for polynomials.

On a human scale the space we inhabit is flat 3+1 dimensional spacetime and many physics phenomena can be well approximated linearly. On a quantum scale the wavefunction seems to be completely linear.

So the answer is that the phenomena you listed are free. If you have a two eggs and add one more you are not going to get zero eggs. Same applies to length, duration, etc.

As to why things are free, hard to say. Some of it might have to do with the locality of the universe (lack of global structure).

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The so-called real numbers are based on the natural numbers, which in turn are based on the primitive notions of first and next as follows:

  1. Let '1' symbolize the first object in some collection of objects.
  2. Every object in that collection has a unique next object. (For convenience, we assume there is no "last" object in that collection.)
  3. The first object (1) is not the next after any object in that collection.
  4. Any pair of different objects in that collection have different next objects from one another.
  5. Every object but the first in that collection can be reached by repeatedly going from one object to the next starting a the first object.

These essential properties of first and next correspond to Peano's Axioms for the natural numbers.

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  • +1 For pointing out that our arithmetic faculties necessitate no one type of object implying they are absolutely necessary to perceiving all types of objects.
    – J D
    Oct 1, 2023 at 16:40
  • Algebraically, it is quite a leap from natural numbers to real numbers. In any case, it is unclear which numbers the OP refers to.
    – nielsen
    Oct 2, 2023 at 12:00
  • Re: "quite a leap from natural numbers to real numbers" It is a gigantic undertaking, but it is apparently possible to formally construct, i.e. prove the existence of the integers, rational and real numbers starting from Peano's Axioms using only basic set theory and the ordinary rules of logic. Oct 2, 2023 at 15:17
  • @DanChristensen That is true for integers and rational numbers. With real numbers, it gets more difficult. See e.g. math.stackexchange.com/questions/652312/….
    – nielsen
    Oct 3, 2023 at 8:58
  • @nielsen It is also true for the real numbers, every real number being a set of rational numbers (Dedekind cuts). Oct 3, 2023 at 16:03
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None of the other answers seem to have mentioned a distinction that may help understand why "numbers apply to such disparate concepts". This is the distinction familiar to logicians but somehow not mentioned often enough in logic textbooks. The distinction is between meta-language natural numbers and object-language natural numbers. While the latter (after adding the negatives) form a "group" as you mentioned (namely, a group under addition), the former are a kind of sorites-like collection that is not necessarily a group. What is actually applicable to "disparate concepts" are the meta-language numbers, which can also be described as the naive counting numbers, etc. They would naturally apply in situations involving any quantitative analysis, which can certainly be "disparate".

The formal mathematical theory, of course, involves the object-language natural numbers, with form a semiring with its familiar properties. Emphasizing the meta-language numbers as an intermediary between the phenomena and the formal theory may help explain their applicability.

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