Mathematical Platonism. Are numbers real?

Question

Often heard this being asked: Are numbers real?

As an answer I offer my own analysis for what its worth.

The color green is considered real. As per scientists it's only distinguishing quality is that it has a wavelength of 555 nm. In essence we're seeing 555 nm; if green is real, so is the number 555 (nm).

Likewise, when a needle pricks me, the pain is nothing more than (say) 5 Newtons of force. If pain is real, so is 5 (Newtons).

What about temperature? The warmth you feel when you stand out in the sun is your skin sensing infrared light; warmth = 1300 (nm), the wavelength of intermediate infrared light. If heat is real then so is the number 1300 (nm).

Basically, our sense organs can and do perceive numbers as sensations.

Are numbers real?

EDIT START

I'm not seeking an explanation for the different classes of numbers (naturals, wholes, fractions, etc.) and nor do I deny that numbers are tools. The received wisdom on numbers is that they're abstract and that's one reason why someone would take the stance that numbers are not real, at least not as real as a mouthful of steak which you can see, touch, smell, taste, and hear. Someone once told me that numbers are not tangible and hence they're not steak-like real and in this question I explain that that may not be completely true. Sensations are quantities (feedback indicates that units of measurment matter when it comes to which number one is sensing - 3 pounds = 1.4 kg)

EDIT END

Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/142072/discussion-on-question-by-agent-smith-mathematical-platonism-are-numbers-real). — Geoffrey Thomas, Jan 15 '23 at 10:45
Logic, ergo Math and numbers, are part of metaphysics, that means they are imaginary, not real. They are not part of physics. See Kant's Critique of Pure Reason, specifically, the _Transcendental Logic_ (Pure General Logic). — RodolfoAP, Jan 16 '23 at 03:39
@RodolfoAP, metaphysics, true - *ontology* deals with *what exists/is real*. — Agent Smith, Jan 16 '23 at 03:44
In order to use nm to assert numbers are real, you might want to support the implication that nm are real. If nm are not real, it casts serious doubt of the realness of numbers. Also, I don’t think the link between pain and 5 is well established here. — Todd Wilcox, Jan 16 '23 at 13:42
Secondary question: are "imaginary" numbers real? If not, and "real" numbers are real, then how can two concepts that do not differ mathematically differ in their reality? — Todd Wilcox, Jan 16 '23 at 13:48
Green is a color, not only a wave length. Yellow and blue combine to make green. Numbers are abstract, not physical. — David Smith, Jan 16 '23 at 23:48
Does this answer your question? [Are numbers real?](https://philosophy.stackexchange.com/questions/8512/are-numbers-real) — Todd Wilcox, Jan 17 '23 at 04:13
Numbers are not necessarily *real* - they can be also integer, rational, complex... just kidding :) — Roger Vadim, Jan 17 '23 at 10:10
@DavidSmith Yellow and blue only combine to make green in our minds based on how our eyes are wired and brain interprets the signals. 3 types of cones peak sensitivity of 564-580nm (yellowish), 534-545nm (green), and 420-440nm (slightly purple to blue) correspond to our primary colors. Different numbers and positions of peaks change which colors are primary. Mantis shrimp have 12 primary colors. — David S, Jan 17 '23 at 18:53
If you assume numbers are 'real' because you can measure things and get those numbers, does that mean that the square root of 2 is not 'real'? — JimmyJames, Jan 17 '23 at 19:07
Tough question @JimmyJames, but the best response to that seems to be *not all* numbers are *perceivable*. Secondly, I'm not claiming numbers have to be perceivable to be real. I'm saying if perception is our criterion for existence, then because we can perceive numbers, numbers are real. — Agent Smith, Jan 17 '23 at 19:11
I guess I would question the assertion that we perceive numbers. We perceive measurements and assign numbers to them. But that aside, are you saying because some numbers are real, all numbers are real? Keep in mind that the numbers most people deal with are an infinitesimal fraction of all numbers. — JimmyJames, Jan 17 '23 at 19:37
@JimmyJames, numbers, as far as I can tell, are all sums of n 1's. So 4 = 1 + 1 + 1 + 1 (4 × 1). If I can ahow that I feel 1, surely I can feel 4 even if no opportunity arises to do so. — Agent Smith, Jan 18 '23 at 06:39
I think you are talking about counting, not numbers. There are many numbers that are not the sum of ones. I already mentioned one that you acknowledged. — JimmyJames, Jan 18 '23 at 14:27
@JimmyJames, you're right and I don't have the requisite information to comment further; suffice it to say that *some* numbers can be *sensed*. It's also true that the *Pythagoreans*, the peeps who claimed *all is number*, were dismayed to find out the truth about the square root of 2. — Agent Smith, Jan 18 '23 at 15:10
I really don't see how you can 'sense' numbers. If a raindrop falls on your head, and splits into a 12 smaller droplets, is that one, 12, or the total number of molecules in the raindrop that you have sensed? If no one is counting, there is no count. — JimmyJames, Jan 18 '23 at 15:48
@JimmyJames, my best answer: If *one* drop falls on your head, you *sense* 1 and if it splits into 12 smaller drops and they all fall on your skin and you feel it, you sense *12* and if you feel the drop's *weight* you're sensing the combined mass of all the molecules in that drop. It's again to do with *units* of measurement (big drops, small drops, molecules). I've tried to come up with a satisfactory response to this objection, but none seem to make the cut. — Agent Smith, Jan 18 '23 at 16:15
Another thing to consider is that 555 nm is not the exact wavelength (google says it's 510 BTW) of green light. That is an arbitrary precision. There's no such thing as a perfectly precise measurement and each measurement will give slightly different answers. We then use statistical methods to come up with a single answer, slice off the decimal places and call it a whole number. Lots of steps involved in sensing that number, which of course is not something our eyes even do. Our eyes detect green without using numbers. — JimmyJames, Jan 18 '23 at 17:58
You have to watch Darren's Aronofsky film "π" about π number, something like this. — άνθρωπος, Jan 30 '23 at 11:31

eigengrau · Accepted Answer · 2023-01-15T01:39:12.393

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By "real" here I assume, by your example, that you're talking about "physically real". And in that case real=experimentally_measurable. And that, in turn, means units. Even your own example uses green=555nm. And 555nm is indeed measurable, as would be 555kg or 555secs, etc. Moreover, the kind of units indicate the kind of experimental apparatus needed to perform the corresponding measurement; length(nm), mass(kg), time(secs) are each measured differently. But 555 with no units at all is not measurable at all, not by any kind of apparatus, hence unambiguously not physically real. But there are other senses/connotations of "real" where you could certainly characterize numbers as that kind of "real".

Now, you could further ask whether a dimensionless ratio, like the fine_structure_constant=1/137, of two measurable quantities, both with the same units, is itself "real". Not sure how to answer that one.

edited Jan 15 '23 at 01:39

answered Jan 15 '23 at 01:29

eigengrau

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I understand. The actual number itself changes with the unit of measurement; nevertheless it *is* a *quantitiy* that's being *measured* by our *senses* . I also assume the universe uses a *system of units* that possess a *rationale* (based on some fundamental numerical values, like for e.g. *Planck units*). Another point worth noting is how the sense organs resemble the Richter scale i.e. if I say my pain went up from 1 to 3, I mean a 1000 fold increase in intensity. Nothing to do with units, but still a very *numerical comparison*. – Agent Smith Jan 15 '23 at 02:21
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@AgentSmith Sure, the human body is a physical system, capable of numerous physical interactions, and can absolutely be considered an "experimental measuring apparatus". But pain has units, not going from 1 to 3, but from, say, 1**ouch** to 3**ouches**. For example, we could externally measure pain with an electric-shock-administering device whose needle is calibrated in, say, **volts**, with a corresponding **ouch** value printed alongside. Then we administer increasing shock levels, and ask you when the two pains are equal. Or maybe an eeg could measure brain activity corresponding to pain. – eigengrau Jan 15 '23 at 03:52
This is right, but you can express it in a simpler way: using numbers to prove that numbers exist is a circular argument, a logical tautology. – RodolfoAP Jan 15 '23 at 05:05
@AgentSmith Simply by talkting abour measurements, you give numbers at least as much reality as you give to the concept of measuring. – Bobby J Jan 15 '23 at 13:04
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555nm == green is a convention based on the average peak sensitivity of one type of human cone cell, in humans that *have* the "normal" set of three types of cones. See https://en.wikipedia.org/wiki/Color_vision. – chepner Jan 15 '23 at 15:28
@BobbyJ, that is correct. We regard heat, force, light, as real but these are actually *quantities* or numbers. – Agent Smith Jan 16 '23 at 03:05
The human interpretation of a physical phenomenon can be a number, but that is a subjective representation, the representation (the number) is not out there. You are saying that political frontiers are physical because you see them on a map. – RodolfoAP Jan 16 '23 at 03:57
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_"But 555 with no units at all is not measurable at all"_ The physicist in me would disagree. There's no way to directly measure something with units. If we use a one-meter long stick to measure a road to be 10 meters wide, we are measuring "10" and using our knowledge of the stick length to say that the width is 10 meters. All measurement devices just measure unitless fractions between quantities of the same dimension, and if one of the quantities is known, we can implicitly measure a quantity with units. But the measurement itself just gives a unitless number. – JiK Jan 16 '23 at 16:57
@Jik, in some context (metric system), green is 555 and so long as you limit yourself to that particular context, you're *sensing* **555**. Plus, the body might possess it's own natural system of units. Lastly, we could say that we experience all the numbers that correspond to 555 in other system of units like e.g. if you eat 4 out of 8 slices of pizza you're also eating 1/2 the pizza. – Agent Smith Jan 17 '23 at 06:08
@AgentSmith The body isn't measuring quantities or numbers. It is reacting to stimuli. For green light, we've given a label of 555nm. Its also 0.00002185 inches. A single needle point of roughly 1.4 quintillion atoms poking your arm has trillions of individual interactions occurring between it and your skin, followed by the cascading reactions each affected cell in your body takes in response. However, those trillions of interactions are all entangled in a manner your brain describes as one poke. In that way I would say dimensionless ratios are more real than unitless numbers. – David S Jan 17 '23 at 19:15
@DavidS, pain is *directly proportional* to the temperature. – Agent Smith Jan 17 '23 at 19:19
@AgentSmith _et al_, Maybe take a look at https://arxiv.org/abs/2110.03514 where the abstract begins, "Physical dimensions are **not numbers**, but used as numbers to perform dimensional analysis...", and see Section 1 for further elaboration. I think maybe that might help clarify the distinction my answer tries to emphasize by "real=experimentally_measurable". You guys seem to be wandering into different senses/connotations of "real", where the answer already acknowledges that numbers are definitely "real" in some senses (but definitely not the canonical physical sense) of the word. – eigengrau Jan 18 '23 at 04:29
@eigengrau. I replied to another poster that *pain* is *directly proportional* to *temperature*. – Agent Smith Jan 18 '23 at 06:42

score 10 · Answer 2 · answered Jan 14 '23 at 22:06

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Asking whether a number, such as four, is real is like asking whether a word such as 'big' is real. The qualities which we think of as big are real. When we say a football stadium, for example, is big, we are referring to the size of the football stadium, which is real. The quantities which we think of as four- such as four shoes or four cars- are real. The number four, however, is just a token we use to denote such quantities, just as the word big is just a token we use to denote sizes.

answered Jan 14 '23 at 22:06

Marco Ocram

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Agreed. Numbers are not in nature. Numbers are constructions on the subject side. And actually, more precision would be warranted. Natural numbers would be easier to claim "in nature" as reals, which are clearly a human construction only (to wit, infinite number of decimals are not acceptable in a finite amount of space/information). So "numbers are real" needs to be refined before it actually makes sense. – Frank Jan 14 '23 at 22:44
To further enrich my position I would say we could *sense* the number 2, not as 2, but as *twice*. You'll feel the "difference" between a coke can 2 coke cans as 2 of them being *heavier* (twice the weight). – Agent Smith Jan 15 '23 at 02:30
Not sure this is going anywhere. Some platonists in antiquity actually worshiped some numbers, apparently. Feel free. – Frank Jan 15 '23 at 03:18
@Frank, . Are you perchance referring to the *Pythagoreans*? – Agent Smith Jan 15 '23 at 03:52
Ah yes - must have been them. – Frank Jan 15 '23 at 04:17
"When we say a football stadium, for example, is big, we are referring to the size of the football stadium, which is real." @marco-ocram I beg to differ: we are stating an opinion about the size of the stadium. A person who grew up in a hamlet may say that a 2000 seat stadium is big, because it it bigger than his entire hamlet; but someone else who is used to the Melbourne Cricket Ground (capacity 100,000) would see it differently. – Simon Crase Jan 15 '23 at 21:03

Hank Igoe · Answer 3 · 2023-01-17T00:42:59.697

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Mathematicians, specifically set theorists, have so little faith in the existence of numbers that they must posit an axiom for something even as fundamentally obvious as the existence of an empty set.

EDIT: In light of Peter's comment below, and on a less flippant note, consider the Church numerals. When one calls the numeral/function three on the function (technically, procedure) "Print 'hello world'", the terminal will show:

hello world
hello world
hello world

But note that they are called "Church numerals" rather than "Church numbers." Whether the number three which describes the cardinality of that set objectively exists is debatable. But certainly the numeral does.

edited Jan 17 '23 at 00:42

answered Jan 15 '23 at 11:05

Hank Igoe

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It's the easy way out. Avoid controversy at all costs otherwise they'll never get anything off the ground. The question of whether numbers exist is philosophical, working with them is mathematics. – Agent Smith Jan 16 '23 at 02:46
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You would use axioms regardless of whether you believed in something or not. – kutschkem Jan 16 '23 at 08:32
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As a mathematical logician, this answer makes no sense to me — how on earth does the axiom for the empty set imply any lack of faith in the existence of numbers? A big motivation for the axiomatic approach is that it’s compatible with a wide range of philosophical views — from a pure formalist who sees axioms as just rules of a meaningless game, to a convinced platonist for whom the axioms express self-evident basic truths, via (e.g.) a pluralist or skeptical platonist who believes some mathematical reality exists but takes these particular axioms as hypotheticals. – Peter LeFanu Lumsdaine Jan 16 '23 at 22:46
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You got it the other way around: mathematicians believe *so much* in the empty set that they have no problem agreeing on its existence without proof, ie giving its existence as an axiom. When a mathematician is not conviced of a fact, they either need a proof of it or they just don't consider it given. – seldon Jan 17 '23 at 12:41

Chris Degnen · Answer 4 · 2023-01-15T00:20:54.923

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An interesting number like e, Euler's number, a 'constant of nature', as real as could be. It is known inexhaustively by many representations. Many discoveries and many perspectives, but never the complete picture. The abyssal ground of e is perhaps as endless as e itself. What we have are only the pieces we know about an unknown proto-e we do not entirely know. What is known as e is the real e as much as it is a real idea and 'concept' or collection of discoveries, but it cannot exhaustively be the true e, which likely the OP may have considered as the real e. No doubt the true e is without our determinations, furthermore.

This reduction is a tougher sell with the number 1 but I assume the same principle applies.

edited Jan 15 '23 at 00:20

answered Jan 14 '23 at 23:37

Chris Degnen

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You are on to something - exponentials are a boon, for many reasons. Very easy to work with, showing up in all sorts of places like trigonometry and Lie algebras, matrix exponentials ... very useful little function and so well behaved. – Frank Jan 15 '23 at 00:38
There can be a force = ***e*** and then we could *feel* it, preferrably not as a punch or a kick. – Agent Smith Jan 15 '23 at 02:26
Actually, there couldn't be. Since e has infinitely decimals, but nothing physical has enough capacity for an infinite amount of information? – Frank Jan 15 '23 at 03:16
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Well then *feeling* the *pressures* of *population explosion* could be taken as a psychosomatic measurement of *e*. – Agent Smith Jan 15 '23 at 04:10
Agent Smith: There could be a force of `e` Newtons, or `e` pounds, or whatever. But as @Frank says, you couldn't measure it accurately enough to be sure it actually was `e` in whatever units you measured in. Most physical effects are continuous (classically at least; space as well as mass might actually be quantized) and thus can have any real value. (e.g. electrostatic repulsion force scales with 1/r^2 so you can make pretty fine adjustments.) The distance and/or the charge, and/or the electrostatic constant, would have to be irrational, though, since `e` is irrational. – Peter Cordes Jan 15 '23 at 04:52

RodolfoAP · Answer 5 · 2023-01-15T05:38:27.373

The question assumes multiple facts that are not, the language is imprecise, and some elements are incorrect. This is moreover a long comment.

Real in this context refers to the counterpart of imaginary. That is, moreover, physical, not subjective, that is, objective (that it exists independently of humans, that means, even if humans would not). How can a number be real, when it has such quality?
From a philosophical standpoint, numbers are generals, not particulars, they designate a single quality of a set of particulars: its accountability. You can say that a particular "is real", but the general is precisely the opposite: an idea abstracting multiple qualities of a set of particulars. In general, particulars have physical substance, generals don't.
Green is not "real". Green is a subjective perception. Some types of color blindness don't allow to see green. Ergo, green is subjective.
In order to prove that numbers exist, you associate green with a frequency/period, which depends on numbers. Your argument is then that numbers are real because numbers prove it, that is a circular argument. You are trying to prove the reality of something that you implicitly assume to be real.
In order to count something, you need of real (physical, objective) limits, borders (how would you count circles drawn in a paper, if circles would not have borders?). The key of your question is here.
- Consider a rainbow. A rainbow does not exist in a precise part of space: an observer in a different position sees it in a different location; so, its borders are subjective. If you touch it with a finger the size of the galaxy, it will move as a whole. But internally, there's just atoms that can be perceived according to the subject biases.
- An apple or a rock are exactly the same thing, at multiple different scales: they are less fragile to dissipation, they are more compact than a rainbow, they are formed of different elements. But besides such different qualia, a rainbow and a rock are exactly the same: things with borders we create, that seem real, physical, but are largely dependent on the observer.
- The only thing that makes you believe that a rock exists outside of your body and a rainbow doesn't, is your subjective biases, physical and rational. A Martian the size of a quark will not be able to know or touch an apple, he'll just perceive fields. And a galactic giant might feel the weight of a rainbow in its finger.
- So, rainbows, rocks, waves, fields, dark matter, whatever, is just a rational construct that depend on your subjective potentials.
So, strictly, nothing is real, objective, everything (specifically, the "thing" part) is subjective.
If you need to split nature in two parts, the real and the non-real, you will need a quite precise definition of real. Otherwise, everything is just a mess of energy types we can interact with.
Anyway, numbers, in all senses, are subjective (not only in the subjectivities that define what can be numbered). Numbers are ideas. Numbers are represented by symbols, that are subjective to each culture. Numbers are organized in numbering systems, and all are valid to count, etc.

As a general rule, I would express it this way (from a text of mine): "The object is the interactional counterpart of the subject. None exists without the other. The subject determines the object in its totality." So, when you think of yourself, you interact with yourself, you are acting as a subject and an object, in order to exist, either physically or rationally. You see? Simpler. That is the precise sense of cogito ergo sum.

I'm intrigued by your comment that *color* is *not* real. Color is a *specific wavelength* of *light* oui? Set that aside for the moment. Is *mass* and the corresponding *weight* real? — Agent Smith, Jan 15 '23 at 06:29
But the numbers e and tau (or 2*pi), for instance are not subjective; nor, indeed are a lot of mathematical concepts, not to mention dimensionless physical constants like the fine-structure constant. No matter how one *expresses* the numbers, and no matter the size of the observer, e will remain the unique a for which d/dx a^x = a^x, the ratio of a circle's circumference to its radius will be tau, 107 will be prime, etc. Those are objectively true. — Eric Snyder, Jan 15 '23 at 08:30
Your call for a *precise definition* of *real* is on point. That's why I was careful: *If x is real, so is y* format. — Agent Smith, Jan 16 '23 at 02:58
@EricSnyder constants like pi or e are representations, not facts. Pi depends on the ideal of a circle, circles are not physical (we can perceive a circle, but since everything is made of particles, it will not be circular, so, where you perceive circles, I perceive polygons, any my value of pi is larger than yours). _pi_ or _e_ depend on quantities, that is, numbers; so, based on numbers, you are trying to prove the reality of numbers; that is a circular argument: "numbers exist because the relation of this 2 numbers is always the same". — RodolfoAP, Jan 16 '23 at 04:29
"I'm intrigued by your comment that color is not real. Color is a specific wavelength of light oui?" No, because the same color can be created by different combinations of wavelengths — by a pure wavelength, e.g., 555nm, but also by combinations of different wavelengths. It's the same subjective experience, and our eye can't distinguish. Colors are created in our minds as responses to "real" electromagnetic radiation. — Chris Bouchard, Jan 17 '23 at 13:54
OK, looking back at your original answer, perhaps we're talking past one another. I'm suggesting that the axiom *The universe around us is real* eventually produces numbers as a result. But you say *"So, strictly, nothing is real, objective, everything (specifically, the "thing" part) is subjective."* ::: Can we agree that the reality of the universe you and I both inhabit is axiomatic? We literally can't have a conversation without that starting point, as we must conclude from its lack that you, me, and the idea of conversation are not real. — Eric Snyder, Jan 18 '23 at 11:24
@EricSnyder when you say "reality is axiomatic", you mean that it is apodictic, unquestionable, and so, you assume that I also agree with physical realism, which I don't, I only find consistent the Kantian position: _knowing the noumenon_ (which is essentially your axiomatic reality) _is not possible_. See Immanuel Kant, David Hume, George Berkeley. And precisely there lies your issue: if we can't know the noumenon, how can we accept that it is accountable? Conversely, if it is accountable, the noumenon becomes identical to phenomena, and with trivial proof we know it cannot be. — RodolfoAP, Jan 18 '23 at 12:15

score 1 · Answer 6 · answered Jan 15 '23 at 15:22

This is my first post here. I cannot comment yet.

In the movie matrix the character morpheus asked "How would you define reality? Is it what your senses feel?" or something like that.

I think there is an objective reality. Sun existed long before there were any humans. We have evidence for that.

However, we cannot know about its existence if none of its effects passed our senses. Consider a blind man going out in open when cool air is blowing. He feel no effect of sun so he cannot tell its day outside or night.

If we take that as definition of how we know something is real we have to say that no, numbers are not real. Idea of numbers, sure, but not numbers.

Its like idea of vacuum. Nobody ever encountered real vacuum. Its just in head. Its an idea. Its like superman.

Idealism does ness up me argument. Hence the format I used: if sensations are real, so are numbers because sensations = numbers. — Agent Smith, Jan 16 '23 at 03:03

score 1 · Answer 7 · edited Jan 30 '23 at 09:12

The underlying question is - can the components of mathematics (like numbers) be separated from the processes of computation?

Take pi. Pi is a number, pure and simple. If pi can not considered to be a number, then nothing can be a number. If pi "exists" on its own, does it have the properties that we ascribe to other obviously real things? The answer is "no" for some of the things we can explain physically, like mass, size, shape, texture, odor & color.

Yet it is computationally derived in many, many ways and the same result is always obtained. Pi is reproducible, testable, invariant and unique. It has definable properties, like other things that we would recognize to be real. Is having definable and testable properties part of the requirements for some "thing" to be real? That's worth thinking about. Maybe.

Yet again, Pi is irrational - no computational sequence can define it exactly and it is felt on solid grounds to be undefinable in a perfect way by any computational process. Its digits go on forever. That doesn't make it sound real at first. If we know about it only because of computation and computation can't exactly define it, we are faced with a question. Does this prove that Pi is not real?

Yet again [again], Werner Heisenberg and a century of physicists following him come to the rescue. They tell us that nothing in the very real physical world can have its properties physically defined in an exact way, either.

Sadly without the concept of reality, our tiny little biological minds stop working. We need it, whether or not the universe needs it. We need to know if the components of math (like numbers) are real. So can physics guide us?

A photon in flight from a distant galaxy has no defined physical properties until it is "observed" (that has nothing to do with our senses, it refers to the photon interacting with something.) We can definitively show that its behavior is consistent with a mathematically defined "wave packet", although there is no cosmic computer doing the wave packet's calculations along the way on the long trip from Andromeda to our telescope's sensor. According to relativity, zero time elapses during the trip from the standpoint of a photon traveling at the speed of light in a vacuum. Even if such a cosmic computer wanted to, there is no time for calculations, at least the way we can imagine doing calculations. Yet again, we know intuitively that the photon is real in some way during the trip. On one hand, it is very real and we can show through experimentation that it is a mathematically described entity, but on the other hand mathematics doesn't have time to compute it.

So yeah - we make a leap of faith and conclude that photons between galaxies are somehow "real". Pi is real also, even when it has not been calculated by some computer or sequentially derived by some math professor in one of a zillion different ways. So is pi squared real, the base 10 logarithm of 4,351,199 is real and so is any other number.

We know this because we assert that the universe is real and the real things in the universe clearly are described by mathematical rules without a Matrix style computer doing all the computations. And we don't know how to understand math without the numbers and the temporally sequential steps that go into mathematical reasoning and computation. So as far as we can tell, math just IS. Numbers just ARE. Those are just variants of the verb "to be, which means "to exist"."If something exists, then it is real. Maybe Neo would have a different viewpoint.

Your point is a good one. Assuming our senses are *computing* the value of *numbers* being perceived, if numbers like *pi* or *e* can't be computed, then they *can't be* perceived. However, there are some *cheat codes* - doing a U turn (180 degrees) is to *experience*, in radians, *pi*. — Agent Smith, Jan 16 '23 at 02:55
I feel obligated to point out that Pi is very much a [computable number](https://en.wikipedia.org/wiki/Computable_number). You can calculate just as many digits as you like with any of several well-known formulas, and you can take any of those formulas to be the definition of Pi, if desired, or you can use the standard geometric definition. On the other hand, there are also "real" numbers which are genuinely uncomputable, and they are in fact the vast majority of "real" numbers. — Kevin, Jan 18 '23 at 00:58

score 1 · Answer 8 · answered Jan 16 '23 at 11:06

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This is an ill-formed question because you don't explain what you mean by "real." And I'm not even sure you have a good idea of what you mean by "numbers" (though that may not be quite so important to this question).

Numbers are certainly a concept that people think about in many ways, and quite some work has been put into deep and logical elucidations of what numbers are. (And there are many of these, depending on the axioms you start with. Even a simple question such as, "does 2 + 2 = 4" has different answers, depending on the number system you're using. In many variants of the everyday systems you use, 2 + 2 = 4 is indeed true. But in modular arithmetic, modulo 4, 2 + 2 = 0. The basic idea underlying "numbers" does not need numbers at all.¹)

You might consider other concepts and see whether you think they are "real." For example, is "love" real? It's not something you can measure numerically, and often not even comparable: does Alice love Bob more than Bob loves Charles?²

Yet, if Alice murders Charles because (she says) she loves Bob and can't stand that Bob loves Charles instead of her, that's a pretty dramatic real-world effect for a mere concept we can't measure or even clearly define.

^{¹ This framing of "numbers" as 0, 1, 2, 3, ⋯ causes enough trouble and confusion that it's quite common in mathematics to drop all that and instead use a Peano system with just a "zero" and a successor function: ∅, S(∅), S(S(∅)), ⋯. Note that "∅" here is not necessarily the same thing as the number "0"; the above works just as well if you define the first natural number to be "1," and many do.}

^{² And these conceptual things can even sometimes comparable and sometimes not: consider a poset of { x, y, z } where y > x and z > x may both be true, but you cannot compare y and z: both y > z and z > y are neither true nor false; they statements as invalid as "+ = 4 2 3."}

answered Jan 16 '23 at 11:06

cjs

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True, true, and please read the other answers and my reply to them. – Agent Smith Jan 16 '23 at 11:54
@AgentSmith I have read your comments on other answers, and I see the same issues I described here. The primary issue is having no good definition (or definition at all, as far as I can tell) of what you mean by "real." A secondary issue is what I feel is a particularly limited view of "numbers," such as your "sensations = numbers" comment. (That you can _sometimes_ connect numbers and sensations does not mean that any sensations require numbers or that all applications of numbers to things induce sensations.) – cjs Jan 16 '23 at 12:54
objection sustained. Imagine you hold out your hand, palm facing me. I push against that palm with my fist, gently at first and then harder. You sense *something* increasing. That *something* if it is *not* a *quantity* (describable only numerically), what is it? – Agent Smith Jan 16 '23 at 13:10
@AgentSmith It's merely a sensation. Consider my example of love, which is also a sensation, and pretty clearly has real-world effects. How would you describe love numerically? On the other side, consider the [Gödel numbers](https://en.wikipedia.org/wiki/G%C3%B6del_numbering) you can assign to well-formed formulae. What sensation do those produce? To what real-world objects do they even map? (The WFFs appear to me to be no more—and possibly even less—"real" than their numbers: what is the reality of "∀x ∀x′ (x+x′ = x′+x)" ("addition is commutative")?) – cjs Jan 16 '23 at 14:13
Unless otherwise stayed I’m pretty sure it’s standard that “realism” in math means platonically existing. – J Kusin Jan 16 '23 at 17:13
@JKusin Well, Wikipedia says, "Mathematical Platonism is the form of [mathematical] realism that suggests that mathematical entities are abstract [and] have no spatiotemporal or causal properties...." Which seems very different from what the OP is talking about as "real" in the question and his comments, where he focuses very much on spatiotemporal attributes such as colour, force and temperature. – cjs Jan 16 '23 at 18:53
@cjs I took it to mean, are there any physical facts like wavelength of EM which could make us believe numbers are platonic (real). Quine gave a way for this; that numbers as platonically construed by most mathematicians are indispensable to science, and therefore exist platonically. – J Kusin Jan 16 '23 at 20:35
@JKusin That would certainly seem to lead towards a clarification of the question, but hardly a complete one. Can one map each of the infinite natural numbers on to a physical element in the universe? If not, does that mean that the first _n_ natural numbers are "real" and the remainder are not? If that's the case, are natural numbers "real" or not? (And of course there are many other types of numbers with which one can play this game. It might be useful to start with finding examples of numbers that are clearly not "real" by whatever definition is intended.) – cjs Jan 17 '23 at 01:58
@cjs, good point. First, is *infinity* a *number*? If it's only a iterative procedure (algorithmic loop) we experience that everyday (lather, rinse, *repeat*). Second, we may experience *relative infinity* - to a child who has learned to count only up to 10, what is 11? The bone that shatters when the force applied to it exceeds its tensile strength is, in a sense, infinity. Thirdly, as you already know 1 + 2 + 3 + ... = -1/12 [kind courtesy Euler]. Fourthly, the mathematics of infinity is a work in progress. We shouldn't that is to say jump the gun. – Agent Smith Jan 17 '23 at 05:43
@AgentSmith _All_ mathematics is a work in progress to some degree or other. But in some areas we have quite solid systems we can use without worrying about them changing under us. Many of the various definitions natural numbers, for example. ¶ To understand what you mean by "number," we need your definitions. (Is "0" a natural number? By some definitions yes, by others no. Both are fine, and both definitions are useful. Does a triangle have inside angles that sum to 180°? It depends on the axioms you've chosen for your geometry. Again, each way of doing it is fine and has it uses.) – cjs Jan 17 '23 at 07:48
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As for whether "infinity" is a number, there's no need to worry about that. What I raised is different: by most definitions of "natural number" there are an _infinite number_ of them (none of which is "infinity"). It seems to me that if for a number to be "real" you must have in some way a function mapping it to something in the universe, then not all natural numbers can be "real" unless the universe is also infinite. In which case, _which_ numbers are "real" now depends on your mapping. – cjs Jan 17 '23 at 07:51
@cjs, excellent point mon ami! You hit the nail on the head. However, I wouldn't say "emotions are unreal just because I haven't experienced *all* of them?" – Agent Smith Jan 17 '23 at 09:26
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@AgentSmith If you're talking about the "infinite natural numbers" issue, it's not about "you didn't experience"; it's about "you _can't_ experience." If the universe is finite, and you go around assigning a natural number to everything in the universe (for pretty much any definition of "everything" you like), at some point you'll have assigned a number to everything and you'll still have more natural numbers left over. Those don't and _cannot_ correspond to anything in the universe because there's nothing left for them to correspond to. – cjs Jan 17 '23 at 10:05

score 1 · Answer 9 · answered Jan 17 '23 at 01:50

I would actually like to echo @JiK a bit, especially since I am a physicist.

One of my first lessons in physics was on the fundamental concept of measure and measurement. We we tasked with defining and coming up with our own concept of length. We split into groups and those of us who paid attention to the lecture found a convenient object to define as a reference unit. Some chose pens or markers. Others chose erasers. Some didn't choose anything. One student who must've had a parent who studied physics chose a flashlight and a stopwatch.

Once chosen we were each tasked with measuring a length on the white-board without disturbing or accidentally smudging it and the figure out how to convert from group-A's pen-units to group-B's eraser-units. The group with the flash-light could not proceed, lacking the precision required to measure nano-seconds on their stop-watch.

To find the conversion rates, the goal is to then go to each group and have them make a 'number-line' where each tick-mark on the line represents one unit of eraser/pen/marker/coin/etc and carefully aligning the lines and marks to determine that 10 erasers is 3 pens or 20 coins (as an example, exact numbers must be obtained yourself). To get really accurate conversion ratios, the lines must be drawn out not just on a single paper, but across many. This allows the students to see that in fact 100 eraser-tick-marks can fit 31 pen marks and 199 coin marks. To get 'perfect' measurements of conversion ratios the number of tick marks that must be made and accurately counted becomes endless. In other words, despite the best tools and technologies available to high school students there is no such thing as a perfect ratio.

The teacher then stole one eraser-unit and broke it. Then he asked the class if the device could still produce accurate measurements. Some said no. Some said yes, but you'd have to glue it back together. The teacher then went up to the board and measured the drawn line with the broken pen and counted out the number of ticks that the broken piece could make just as had been done with the eraser before it was split. Of course the tick marks were different, but the broken-eraser-unit still produced a measurement which could be put into ratio with the unbroken-eraser-unit and the pen-unit.

At the end the teacher measured the length of the line in centimeters and converted that to pen-units.

I suppose my take-away was that numbers in the process of measurement are tools, or perhaps a better term is instructions, that tell the reader how to get a specific quantity. A number on it's own doesn't mean anything. There is no length represented by '5' any more so than there is a mass represented by '10'. The same number can represent different lengths because the other half of a physically real measurement is the unit.

Physically real quantities are more than just numbers. They are numbers and a physical-object. A directed-displacement is numbers and vectors. Energy is a number and a unit of work. Delays are a number and a length of time.

Are numbers themselves real? Sure, insofar as a set of instructions are or words in general. But they do not exist independently of people or their perception and understanding. Integers certainly exist in the conception of even the like of bees, ants, birds, dogs, humans, and fish. But fractions (ratios) require another level of abstraction. Not just the ability to count, but the ability to communicate the count and determine what that count means in a different context (covert from Ant-A steps to Ant-B steps). Real numbers require yet more abstraction.

I think the most meaningful answer is a rhetorical question: "Do you know what [ concept ] is?" If you can say yes and accurately use the concept, then yes you think it exists. If you cannot say yes or cannot accurately use it then you do not think it exists. Perhaps the better expression is to say that "The act of creating an idea makes it real for you." Surely the physical distance between object exists independently of humans or anything else, but the number of steps to cross that distance is dependent on people.

Others mention unit-less numbers, but all numbers in physics come from ratios of units. That is they are made from physical objects and physically real distances and masses and delays and energies and ... I don't think the fine structure constant is any more real than any other number, especially since it is a ratio of real quantities, but even more to the point it isn't even exactly 1/137. It is approximately 1/137.

Others point to π or e as being real, but these are still ratios of physical objects. Additionally, drawing a circle on a curved surface increases or decreases the value you get for π depending on the geometry of the surface (positive, negative, or zero curvature). On a sphere the sphere-π (which is obtained from the ratio of the circumference to the radius, not of the sphere but the circle on the sphere) is less than plane-π. On a negatively curved surface, the saddle-π is greater than plane-π. These are all still ratios of lengths.

I will end this thought here.

First off, muchas gracias for your answer. It seems that to specify a number as a sensation is fraught with issues like the one you raise (units). So, how about this then: We experience rise/fall in sensations (objects getting heavier/lighter, a room getting hotter/colder, etc.). These gradations make sense only *quantitatively* and if you can sense them, you sense *numbers*. — Agent Smith, Jan 17 '23 at 04:18
You sense forces. Forces can be expressed in terms of unit, direction and magnitude. There are three necessary components to specify. The electromagnetic force induces forces on your retina allowing sight and heat on your skin while gravitational forces exert forces on your whole body. The ground experts a force on your feet opposing the force of gravity, and is called the normal force. — Gerald, Jan 17 '23 at 05:10

score 0 · Answer 10 · answered Jan 15 '23 at 08:28

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From a programmers point of view

There are only two numbers: zero and one, meaning there is not and there is. This they call a bit, all else is structure.
What programmers do to handle numbers is to call an ordered group of 8 bits a "byte" with the rule "moving a bit one place to the left doubles the represented quantity" and use it to represent numbers 0 to 255. This is just a convention, the rest is left as an exercise...
How you use / interpret numbers depends on your selection of units, e.g. for force: 5 Newtons are 00000101b Newtons. 00000100b becomes 4, plus one gives five. Numbers, like words, are just structured digits / letters / phonemes.
The true philosophical question is now: how turns structure into meaning? How turns sound into speech? How real are words - looking at numbers just as a subset of all words.

answered Jan 15 '23 at 08:28

Bobby J

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Please read my reply to **eigengrau**. – Agent Smith Jan 15 '23 at 09:48
@AgentSmith You want to use numbers for measuring, therefore you also need - as you say - units. By measuring you use numbers for counting (units), comparing (count of units to a value), creating a scale (lin, log. exp) to represent a value by a number. Just counting declares and creates numbers: 1 + 1 = 2, or successor(1) = 2. By accepting that "1+1" has a meaning, you give numbers reality - or accept that arithmetic (applied to units) already has given numbers their reality. – Bobby J Jan 15 '23 at 13:19
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Perhaps we could use [i]ratios[/i] which are as per mathematicians universal e.g. the ratio between 4 inches and 2 inches is the same as the ratio between 10 cm and 5 cm [units cancel out, ratios are dimensionless, pure numbers]. – Agent Smith Jan 15 '23 at 15:27
A ratio is an ideal relationship, not real. But the worst is that a ratio depends on quantities (numbers), so, you try to prove the existence of numbers based on numbers. – RodolfoAP Jan 16 '23 at 04:07

score 0 · Answer 11 · answered Jan 16 '23 at 00:33

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I think something is real if it exists independently of my mind. So in that sense, math is real. 1+1=2 regardless of whether I want it to or not.

answered Jan 16 '23 at 00:33

yters

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It's refreshing to seem people cut through all the noise and zero in on the nub of the issue. Forces, mass, light are considered *mind independent* (excepting idealism). If they are real and if they are essentially numbers, numbers are real as well. Oui? – Agent Smith Jan 16 '23 at 03:01
Math is not physics. Math and numbers are part of metaphysics, that is, _beyond physics_, that is, imaginary, that is, not real. Forces depend on change, which requires of memory and reason, which are subjective, that is, imaginary. Don't take science for granted. The problem of science is the assumption of the possibility of total objectivity, which implies the total exclusion of the subject. – RodolfoAP Jan 16 '23 at 03:32

score 0 · Answer 12 · answered Jan 16 '23 at 08:57

As usual in philosophia, the question if about what we considere to be real. A trivial example that would not make sense for Platon, if the set of real numbers that we learn in school. According to that definition, some numbers are real (integers, rational numbers, and even irrational like pi), and some are not like the imaginary i defined by i * i = -1.

In the context of Platon world, real things exists independantly of human thought: a rock, a plant, or an animal are real things in that context. In the opposite directions, words are not. They are only human conventions first invented to describe real things and later used for general communication between human beings. And they can be used for generalization. For example Platon asked someone to describe what an insect was, and did not accept an enumeration of some insects that were around like a moth, a bee, or a mosquito as a true definition. Of course the bee was a real thing, as was the moth. But the definition of what an insect is is not: it is what we define to be an insect.

Numbers and more generally mathematical concepts are close to words in that perspective. When I see two cows, they are of course real things, as would be two cats of two dogs. That does not imply the number 2 to be real: it is a generalized concepts of one thing added to a similar one, that allow human beings to do more complex operations. Once I know about numbers and arithmetics, if I know the quantity of wheat that an hectare field can produce, how many wheat is required to feed an individual man and the population of the city, I can compute the surface that has to be sowed.

In that sense, numbers are not real. They are of course used and even required to describe real things like the wave length of a monochrome green light. But what a number represents depends of the context where it is used: 555 nm can represent a wave length and 555 cows in a field have nothing to do with that, even if the same number if used.

Interesting post mon ami. I like the comparisons made - insightful. — Agent Smith, Jan 16 '23 at 09:03

Chris Bouchard · Answer 13 · 2023-01-17T14:30:15.910

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I don't think your arguments follow. You make the point that

Likewise, when a needle pricks me, the pain is nothing more than (say) 5 Newtons of force. If pain is real, so is 5 (Newtons).

What about temperature? The warmth you feel when you stand out in the sun is your skin sensing infrared light; warmth = 1300 (nm), the wavelength of intermediate infrared light. If heat is real then so is the number 1300 (nm).

Others have brought up the concept of unit as a way to separate numbers from “reality,” but I would go a step further and say there is no actual connection between what you experience and a number.

Units like newtons and nanometers are not blessed. The force 5 newtons is equivalent to 5 000 000 dynes or approximately 1.12404 pounds-force. Similarly, 1300nm is equivalent to 13 000 angstroms, approximately 5.1181 × 10^-5 inches, 7.0194 × 10^-10 nautical miles, 2.5849 × 10^-6 rods, and 7.527 microChrisBouchards, which is a unit of length I just created based on my height.

They're also equal to 1 force-of-the-needle-on-Agent-Smith's-skin-when-they-got-an-injection-on-such-and-such-date and 1 wavelength-of-the-electromagnetic-wave-that-hit-Agent-Smith's-skin-at-such-and-such-instant, respectively.

There is no actual correspondence between physical values and numbers, because it depends entirely on the units we choose. There's nothing “five-y” about the pain you expernienced, nor “one-thousand-three-hundred-y” about that particular electromagnetic wave that hit your skin. Any number — whole, fractional, or decimal — could be used to describe them, with appropriate choice of units. You pick an arbitrary physical value and a number, and I can give you a unit that makes it describe that physical value.

In this sense, numbers are more like words — a language we've created to describe reality, but that exists only in our minds. We project it onto reality to categorize and understand, but they are constructs. You say 5 newtons, I say 1.12404 pounds-force. You say “pain,” but someone in Argentina says “dolor.” None of it is the physical sensation of feeling pain, just words to describe it based on mental models.

I think the best you can say is that physical values — the things we experience — seem to have dimensions. They seem to have values associated with them of different amplitudes, which we can measure and compare. But we can't really say those values are numbers, because they don't correspond to numbers.

edited Jan 17 '23 at 14:30

answered Jan 17 '23 at 14:25

Chris Bouchard

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I concur, we don't experience a particular numerical value; the best that can be said is given a context (a system of units), we experience so and so number. However, this doesn't mean sensations are *not* quantities. Just because a length can be 1 in one system of units and 2.5 in another doesn't mean we're not dealing with numbers/lengths. Also, I recall commenting that the body may have its own system of units in which sensations take on absolute numerical values and not relative ones. Also, as a side note, Max Planck's units need further investigation. – Agent Smith Jan 17 '23 at 14:52
You're conflating numbers and lengths here. Physical values with dimension could be said to be "real" — e.g., dimensions like length and force — but you haven't established that physical values or their dimensions *are numbers*. It's an abuse of notation to say a length "is 1," and it's hiding the fact that you haven't established this. How can a dimension "be" 1 and also "be" 2.5 if 1 and 2.5 are not themselves equal? "Be 1 in a system of units" is not defined. All we've established is that numbers can be used to *describe* the system, not that they are part of it. – Chris Bouchard Jan 17 '23 at 16:01
well, I'm happy to conflate numbers with lengths, having been taught the *number line* by math teachers in school. Anyway, let's try something different because many posters see this as a flaw in my argument. Suppose there are 2 objects each weighing x units. First I place one of them on my hand ... hmm heavy. Then I place the other too on top ... *heavier*, I'm experiencing/feeling the weight *double* i.e. this has got to be what **2** is. – Agent Smith Jan 17 '23 at 17:21
(Number lines are again not real — only divided lengths of known units, like inch marks on a ruler.) Nothing here *is* 2. It's a physical value whose dimension can be described by an equation *using* two and the dimension of another physical value: *m_combined* = 2 × m_object. This is just another case of units — you're describing the weight of the combined system as 2 unit weights, where the unit is the weight of one object. – Chris Bouchard Jan 17 '23 at 18:15
*Heavier* in my example is *doubling* (× 2) and it doesn't matter what the unit of measurement is - I'm *feeling* the *times 2*, not the weight. – Agent Smith Jan 17 '23 at 19:02
No, you're feeling a weight *x*, and then feeling a weight *y*, and *in your head* relating then by *y* = 2 × *x*. There's no 2 here that's real. – Chris Bouchard Jan 17 '23 at 19:20
My point about units is that "the weight of one of these objects" *is* a unit. You're measuring 2 of those units. But neither one "is" 2 in any meaningful way. – Chris Bouchard Jan 17 '23 at 19:21
The weight has *doubled* and you *sense* it as increased heaviness. – Agent Smith Jan 17 '23 at 19:21
As I said, it doesn't matter what's the unit, even if the weight of the object itself is the unit - *heaviness* is *doubling* or **2** – Agent Smith Jan 17 '23 at 19:23
*You* know that, because doubling exists in your head. But it's just a relation expressed by an equation, not a physical value – Chris Bouchard Jan 17 '23 at 19:23
There's nothing you can point to and say "this is doubling." You can point at one object, then the other, and say "this is double this." But that's a relation, not a value. – Chris Bouchard Jan 17 '23 at 19:24
Last attempt: With your hand, touch two balls. Do you sense **2**? – Agent Smith Jan 17 '23 at 19:24
No, I sense two balls — a physical value, with a unit "balls". It's not 2, I just use 2 in my head to represent it and interpret it. – Chris Bouchard Jan 17 '23 at 19:25
We're discouraged from, as the mods call it, *extended discussions*. My question is, if you're not sensing **2** and it's still about *units* what are you sensing *n* (unit of your choice* of where n is not 2? – Agent Smith Jan 18 '23 at 06:36
I'm not sending any *n*. I'm sensing or measuring *something*, and that something is a physical value with a dimension. *In my mind* I can use all sorts of numbers and equations and formulae to quantify it, but that's the same as using English or Spanish to describe it — it's not *part* of the physical value, but rather something disconnected in my mind that labels it. – Chris Bouchard Jan 18 '23 at 13:28
I will try to update my answer to take all this into account. And to be clear, these dimensions have properties: they have magnitude and they can be compared. But all of this is separate from the numbers that we use in our minds to *model* these physical values, just like they're separate from the English words we use to label them. – Chris Bouchard Jan 18 '23 at 13:31
When I sense or measure something physical, I'm always comparing it to something else physical. I don't ever get the dimension m measuring directly — I get it's relationship to other physical things. There's no direct access. Numbers are a mental tool we've developed to systematize and axiomatize those relationships. We started with tally marks and eventually concluded rules of arithmetic and so on. All those symbols and rules *model* the relationships between physical values, but are not themselves the physical things. They're a language. – Chris Bouchard Jan 18 '23 at 13:38
Ok, mon ami. It was an interesting discussion we had. – Agent Smith Jan 18 '23 at 15:13

score 0 · Answer 14 · answered Jan 17 '23 at 17:45

Reality refers to physical instance where numbers are a representation of reality. In any case a representation can be mingled with reality. It can be accurate, very similar but still different. A reliability of science can be assess by the quality of the representation as regard of reality. You ask if numbers are real? If in nature it is false, but in fact it can be assimilated because our sciences are reliable enough to use numbers to express reality. The real question of your topic is about how far numbers can dive to reflect reality.

Mathematical Platonism. Are numbers real?

14 Answers14

From a programmers point of view