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

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.

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^{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."}

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

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.

Answer 11

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.

Answer 12

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.

Answer 13

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.

Answer 14

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.

Answer 15

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.