Mathematical Platonism. Are numbers real?

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. Commented Jan 15, 2023 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). Commented Jan 16, 2023 at 3:39
• 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? Commented Jan 16, 2023 at 13:48
• Does this answer your question? Are numbers real? Commented Jan 17, 2023 at 4:13
• @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. Commented Jan 18, 2023 at 15:10

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.

• 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. Commented Jan 15, 2023 at 2:21
• @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, 1ouch to 3ouches. 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. Commented Jan 15, 2023 at 3:52
• 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. Commented Jan 15, 2023 at 5:05
• 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 en.wikipedia.org/wiki/Color_vision. Commented Jan 15, 2023 at 15:28
• "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
Commented Jan 16, 2023 at 16:57

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.

• 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. Commented Jan 14, 2023 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). Commented Jan 15, 2023 at 2:30
• Not sure this is going anywhere. Some platonists in antiquity actually worshiped some numbers, apparently. Feel free. Commented Jan 15, 2023 at 3:18
• @Frank, 😄. Are you perchance referring to the Pythagoreans? Commented Jan 15, 2023 at 3:52
• Ah yes - must have been them. Commented Jan 15, 2023 at 4:17

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.

• 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. Commented Jan 16, 2023 at 2:46
• You would use axioms regardless of whether you believed in something or not. Commented Jan 16, 2023 at 8:32
• 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. Commented Jan 16, 2023 at 22:46
• 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. Commented Jan 17, 2023 at 12:41
• @seldon Not all mathematicians do. You don't need to believe in something to use it as an axiom. Commented Aug 6 at 10:45

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.

• 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. Commented Jan 15, 2023 at 0:38
• There can be a force = e and then we could feel it, preferrably not as a punch or a kick. Commented Jan 15, 2023 at 2:26
• Actually, there couldn't be. Since e has infinitely decimals, but nothing physical has enough capacity for an infinite amount of information? Commented Jan 15, 2023 at 3:16
• Well then feeling the pressures of population explosion could be taken as a psychosomatic measurement of e. Commented Jan 15, 2023 at 4:10
• 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. Commented Jan 15, 2023 at 4:52

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? Commented Jan 15, 2023 at 6:29
• 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. Commented Jan 15, 2023 at 8:30
• 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. Commented Jan 16, 2023 at 2:58
• @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". Commented Jan 16, 2023 at 4:29
• "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. Commented Jan 17, 2023 at 13:54

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."

• @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
Commented Jan 16, 2023 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? Commented Jan 16, 2023 at 13:10
• 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
Commented Jan 17, 2023 at 7:51
• @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
Commented Jan 17, 2023 at 10:05

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. Commented Jan 17, 2023 at 4:18
• 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. Commented Jan 17, 2023 at 5:10

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. Commented Jan 16, 2023 at 2:55
• I feel obligated to point out that Pi is very much a 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. Commented Jan 18, 2023 at 0:58
• From Wikipedia: Commented Feb 18, 2023 at 12:20

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.

• @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. Commented Jan 15, 2023 at 13:19
• 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]. Commented Jan 15, 2023 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. Commented Jan 16, 2023 at 4:07

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. Commented Jan 16, 2023 at 3:03

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.

• 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? Commented Jan 16, 2023 at 3:01
• 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. Commented Jan 16, 2023 at 3:32

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. Commented Jan 16, 2023 at 9:03

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.

• 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. Commented Jan 17, 2023 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. Commented Jan 17, 2023 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. Commented Jan 17, 2023 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. Commented Jan 17, 2023 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. Commented Jan 17, 2023 at 19:02

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.

Numbers appear to be abstract. What does this mean?

Firstly, I can not believe I have to say this but, everything that happens or happened is or was real and this is regardless of whether a happening is deemed abstract or not. By real I mean occurring or happening. The phenomenon of thought falls within this definition and so does every phenomenon ever. Again, everything that ever happened is a part of reality, including the phenomenon we call thought. Thus, numbers are real, as is the phenomenon we call consciousness. As a result, I do not believe there is a distinction between the physical and the imaginary. Nonetheless, it seems from my personal experience that numbers exist only within us (intrinsically).

That being said, there are certainly other biological organisms that interpret reality in a conceptually similar way to us in regards to numbers and distinctions. Perhaps although numbers are intrinsic entities or concepts, they are derived from our sensation of the extrinsic world, and therefore, may be a representation of it.

To restate my answer, numbers are real and exists as concepts, within our consciousness which is a apart of reality, which is real.