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I am confused as to what numbers are. Numbers are defined to be what they are, so numbers aren't real? But numbers are found in nature, right? So if we invented them, how can they be found in nature? How can everything be based on something that is not real. Sorry if what I am saying don't make sense, it hard to put it into words. I feel like I've been known numbers all my life yet I don't know what they are. Also why don't complex numbers have an order? Real numbers are an invention just as complex numbers are, so how are imaginary numbers any more imaginary than real numbers? I mean an order was given to the real numbers, so why can't complex numbers be given a definite order?

Thanks

  • Interesting post (Needs more structure though). Just curious, what made you think that numbers are found in nature? (The other day I asked my little brother to go find the number 1. He said "it's easy", went to the clock, and pointed to '1'. Needless to say, I wasn't satisfied with his answer. – Hunan Rostomyan Oct 29 '13 at 5:03
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    When I said numbers are found in nature, I was thinking about how the seeds of sunflowers follow the Fibonacci sequence. – Wobblester Oct 29 '13 at 5:12
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    Excellent. You might find Wittgenstein on rule-following interesting in connection with that thought. In philosophy of mathematics proper, there is the age old problem of the applicability of mathematics; Chris Pincock is great, so check that out. I'm looking forward to seeing how our fellow SEers approach your many interesting questions. – Hunan Rostomyan Oct 29 '13 at 5:26
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    The second part of your question, why complex numbers can't be ordered, is off-topic (it's purely mathematical). You can, of course, define any total ordering you like on it, including ones which are compatible with the total ordering of the real numbers; the problem is that any such ordering will probably not respect algebraic structures within the complex numbers that you might care about. In the parlance, the complex numbers are not an ordered ring. See math.stackexchange.com/questions/181720/… for more details. – Niel de Beaudrap Oct 29 '13 at 11:52

13 Answers 13

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Consider the following analogy. What is a chicken? Are chickens real?

There was a time (most places in Europe, anyway) when this would have seemed an even more stupid question than it does now. Everyone knew exactly what a chicken was. Even a rich noble would have only had to walk perhaps fifteen minutes and point to an example of a chicken. It was a vibrant and noteworthy part of everyone's every-day experience. So too, our experience with numbers. That (point at a carton of six eggs) is six. That (point to an apple, and another apple which has been cut in half and one of the halves removed) is three-halves. And so on.

The fact that you can't just point at a collection of something and say "there is negative-three", "there is square-root-of-five", or "there is six plus three-i" are the reason why some people who are frustrated with those ideas feel justified in saying they aren't actual numbers. It's a fair criticism, in fact, and points to the fact that we never sit down and talk about what numbers are really meant to be. Of course, these days someone could also go their entire lives without seeing a chicken, and they accept that there is an animal which is vaguely involved in the creation of the eggs which they sometimes eat for breakfast. Certainly, for those of us who did not grow up on or near to a farm or a zoo with chickens, we accept the existence of chickens as an article of faith for some years. Similarly do we take the idea there are "numbers" which don't correspond to collections of things as a received idea.

So if numbers don't have to correspond to collections of things, what are they? Well, in the case of (positive) irrational numbers, they can correspond to lengths of lines or of areas — to continuous amounts of something, which is a nice generalisation of collection-sizes. And negative numbers can correspond to deficits or differences of such amounts. And complex numbers, er... well, they're... useful for quantum mechanics, and electrical engineering... and, um, so are quaternions... We find that we stretch the definition of number from "amount of" to "being useful for", which I think is an important thing to notice.

There's no obvious place where we should simply stop. The fact that the complex numbers can't even be ordered any more (never mind the quaternions, for which multiplication doesn't even commute) suggests that just because something solves x²+1=0 does not mean that it's a number (that complex numbers aren't 'numbers' in general). But we can say that just because something is an upper bound for a bounded sequence of numbers, that it isn't a number (real numbers aren't all 'numbers', and the square root of two or five in particular); or that just because something is the difference of two numbers, that it isn't a number (negative numbers aren't all 'numbers'); or that just because something is a ratio of two numbers, doesn't make it a number (positive rational numbers aren't all 'numbers'). But that rules out everything but the non-negative integers; and people have historically even looked askance at zero. You could even argue that one isn't a number, if you argued that by "a number" you mean a plural amount.

So it's pretty important to ask ourselves: what is a number?

What is a chicken? It's a small-ish bird which doesn't fly very well. But we don't want to include kiwis or puffins as 'chickens', so perhaps we should specify that they have short beaks and don't swim well. But what about pheasants? Even if we go on to successfully isolate chickens from all other living birds by definitions, what about the ancestors of chickens who evolved into the modern farm animal? At some point there weren't chickens, and then there were. When did things change?

The problem with chickens, and also with numbers, is that in the end we only have definitions for these words by convention, which are based on examples. We accept modern chickens as 'chickens', and don't accept kiwis as 'chickens'. Similarly, we want to include 'six' and probably 'three-halves' and maybe 'negative-two' and 'square-root-of-five' as numbers, but we don't want to include the function f: ℤ→ℤ given by f(x)=3x+2 as a number. It's not what we want to think of as a number, because it can't be used the way we want to use numbers. Numbers are tools for understanding the world.

Which birds do we accept as chickens? Those that behave in a particular way, and in particular which we can understand in a particular way. Their eggs taste a particular way, their meat tastes a particular way, and they behave in a particular way. We care about how they act and how they taste because we are interested in them as features of the environment which we will interact with (perhaps to eat them). The concept of a chicken is something we have invented to distinguish some animals from others. If we didn't care about the difference between a chicken and a pheasant, we wouldn't have separate ideas for chickens and pheasants. (Just because we have different words for things doesn't make them different, but it does mean that we do care about what differences we think they have.) The concept of 'chicken' is a tool which we use to understand some of the animals we know about.

Similarly, the concept of 'number' is a tool which we use to understand the relationships between objects. But it goes beyond just the concept of 'number' itself: each number is a concept which we use to distinguish from other numbers. We rarely think that there is just "a number" of something, to denote that there is more than zero or one or two; we care about which number. The difference between six eggs and seven eggs is important to us.

But there is another difference with chickens: we may see small chickens or large chickens (a single sort of chicken with different properties), but we never see egg sixes or apple sixes (a single sort of number with different attributes). We do see six eggs or six apples. In this case, the number is not playing the role of a noun, but an adjective. So all this talk of 'chickens', which are objects, has been misleading. What we should have been thinking is something like: "Is red real"? "Is big real"?

Well, colours are real and sizes are real, but what makes a colour 'red'? We can invent an arbitrary definition based on frequencies of light, but then we're making the definition of colour depend on numbers, which is no way to solve the problem of how to understand numbers. In the end we again end up having conventions based on examples. But surely the things which we call numbers must exist? That there really is a number three? We see it all the time, of course. Similarly, there must exist a colour red, mustn't there?

The colour red depends on our sensory apparatus, and the way our brains process the signals sent to us by our eyes. The colour red is an emergent experience, resulting from how our brains and sensory organs are structured. The notion of the colour red is a useful way of understanding our world, based on how we experience it. There's no reasonable way to deny that there are things which shine red light (light which we percieve as red); things which reflect red light (which preferentially reflect light which we percieve as red); and that red light falls roughly within some frequencies of light (we have constructed an entire theoretical apparatus for describing electromagnetism which is useful enough to build radio towers, lightning rods, x-ray machines, NMR machines, and lasers, and in this theory the light which we tend to percieve as red affects certain photosensitive apparatus in a specific way, and these predictions are borne out by experiment). The concept of 'red' is an extremely useful and robust way of describing how we experience the world.

You might even say that the world is described "unreasonably effectively" by the notion of colour; there's no particular reason why so much of our experience should be describable in terms of colour. We don't talk every day about the scent of steel, the sound of plastic, the taste of granite. Somehow, the world is shaped in such a way that our dominant mode of sensory perception happens to be extremely useful for describing a lot of the world. Surely coloured light, in exactly the range of frequencies which we are able to see with our eyes, must play a fundamental role in how the universe works! Surely 'red' has a fundamental reality beyond our own existence; surely the colour red has an unchanging, even Platonic, nature to it!

I disagree. The colour red is indeed a very useful thing to sense, and to understand, because it is how we perceive some useful physical phenomena. But if we percieved a somewhat broader spectrum which included what we call the infra-red, that would be useful too; why don't we? For accidental reasons, I suppose. Perhaps in warm climates there is too much noise in those frequencies; though this doesn't explain why some species of snakes can sense them while we cannot. The reason why we can percieve red among other colours is ultimately because it was a useful accident.

If the number three seems to us to have an extremely vital existence, this could be because the concept of number is a useful one to be able to formulate when reacting to the world around us, and so much so that it is wired into our brains at a very deep level. This means that there are really amounts of things in the world, and that some notions of 'amount' are so simple and important that you can evolve creatures who believe that the notion of amount is so vitally important, that it can exist independently of anything to have an amount of.

The non-negative integers — the "natural numbers" — are just what we call our simplest tools for measuring amount. But they are our tools, extended well beyond our ability to immediately apprehend quantity, off into the dozens, hundreds, and billions — just as we have tools to help us to sense the infra-red, though we cannot directly percieve it.

Numbers are concepts. They are our tools to help us to understand useful things about the world. They are very, very, very useful tools; and versatile enough that we have every reason to believe that they can be used to describe any pattern that we can comprehend (and many that we cannot comprehend) regardless of whether that pattern is ever realized in the material world. But there is no more reason to believe that numbers (such as Three) exist independently, any more than there is to think that there is a Platonic Red which exists independently of any red object.

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    A most excellent answer. +1 – David H Oct 29 '13 at 11:51
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    Piraha tribe that has no concept of numbers would be yet another anthropological confirmation that numbers are man-made tools. – Michael Nov 14 '13 at 1:07
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    what is meant by 'real'? ... without this definition everything is just mumbo-jumbo ;) – slashmais Apr 3 '14 at 6:21
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    This answer isn't as informative as it seems; it begs a whole bunch of questions in the philosophy of mathematics. For instance, the claim that "Numbers are tools for understanding the world" is not at all obvious, and completely ignores positions like mathematical platonism, or intuitionism, or formalism. Moreover claims like "the concept of number is a useful one" are empirical, yet no evidence is provided to back them up. @OP: This is not a good answer. It endorses a particular, controversial, view of numbers. Moreover it doesn't cite any relevant research to back up its claims. – possibleWorld Mar 15 '15 at 4:09
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    @Niel: All that formalism claims is that mathematical objects are certain marks on a page, manipulated according to certain rules (roughly - it will depend on which brand you choose). Importantly, formalists don't think that mathematical expressions express propositions, which is at odds with your claim in the OP that numbers are concepts. Re: the claim that "numbers are useful". I was responding, perhaps not as clearly as I could have, to your quasi-evolutionary argument for some kind of nativism about number concepts. – possibleWorld Mar 15 '15 at 15:47
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The nature of numbers is a real difficult problem; form a "philosophy of mathematics" point of view, the best starting point is yet Frege's Grundlagen (1884 - The Foundations of Arithmetic) - difficult but rewarding. The thorny issue of "reality" of abstract object (starting from Plato and Aristotle) is that we think that objects are real when we are able to see and touch them, and we cannot see and touch numbers. But, if they are not real, why they are so ... useful, indispensable for the entire umankind ? A lot of work in XX century phil of math has been dedicated to find some way to support the idea that numbers are not real (in everyday sense of the term) but mathematics is anyway worth studying as ... a game with symbols, a set of statments true by convention, a social construction, and so on.

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It depends on what exactly you mean by "real". In one view, numbers are just as real as your left hand; they are entities that exist mind-independently, a-causally, and non-spatiotemporally (i.e.outside of space and time). This would be the view of at least one version of mathematical Platonism, and it seems to point to the notion that we are uncovering a deeper and deeper mathematical structure to the universe.

In my view, I would have to say - yes; abstract objects such as the square root of 2 are just as real as a chair, for instance. They are real entities, but they are entities that are not bound by the laws of causation or space and time.

  • Nice answer! It might be interesting to hear a little bit more about why you would recommend your answer here, though. – Joseph Weissman Dec 24 '13 at 19:06
  • your first sentence states the issue, and then you digress... – slashmais Apr 3 '14 at 6:23
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Old question. But fun! I'm surprised no-one mentioned Principia Mathematica wherein over 100 pages (163, if I remember correctly) are dedicated to defining the number "1".

I would play a game, when I was in high school, by suggesting that 2+2=7, and when other students would argue I would simply ask them to prove me wrong. This usually led to a lot of hand hand gestures beginning with 2 fingers plus 2 fingers and usually ending with just one finger.

The summum bonum is simply that numbers are ideas (mental constructs representing a perception, and in that sense, they do exist platonically). As has already been very well explained, these ideas are useful for describing the world around us, and so we continue to use and improve upon these ideas. My suggestion that 2+2=7 breaks the rules outlined by Alfred North Whitehead and Bertrand Russell; but the rules implied by my suggestion are no less arbitrary than theirs, only less useful.

Of course, you should also define "existence" when you ask such a question.

  • does your thoughts exist? what about someone else's (in YOUR context, not the other person's)? – slashmais Apr 3 '14 at 6:29
  • @slashmais Define "exist" and then I'll answer you ;) – Thomas Apr 3 '14 at 19:06
  • I see what u did there :) I tried to point to where I think the answer to a definition of 'exist' can be found here: philosophy.stackexchange.com/a/10552/112 , and in this sense you are completely correct to say that numbers are ideas - everything is. To answer my query about someone else's thoughts: it will 'exist' in your context only when the other person expresses the thought (in/directly) through some behavior that you can become aware of, and from which you can infer such thought. – slashmais Apr 4 '14 at 5:46
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Numbers are "real" in the sense that they are a way that man organizes the relative movement between objects he observes in his environment. (e.g.This here + that there = two of those). However, numbers are not "actual". Meaning that they cannot be qualified as existing apart from the context of objects which man senses. If you remove "number" from the object(s) which give it a definite value, it can only be defined as "infinite". Which, is practically speaking, zero. Thus, numbers, like any abstract concept, require an observer to be "real" (man, in this case). This of course makes the plumb line for ALL value (truth) the one who observes.

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I believe that your confusion is due to not realizing that the "labels" used to categorize the various sets of numbers are just that, labels. The "real" numbers, the "imaginary" numbers, the "complex numbers, etc. are all ordered sets. Unfortunately, some of these labels have other meanings outside of math. Outside of math, "real" usually means something tangible that is perceived by at least one of our senses, and "imaginary" means something intangible and not perceived by our senses. But in math, these words are just labels used to distinguish different sets of numbers. The person(s) that labeled the numbers could have used green instead of "real" and red instead of "imaginary" and we would have the green number set, the red number set, etc.

  • The "only" problem I see in your explanation is this : in what sense reduction of numbers to sets is e real "explanation" ? In what sense are we more confident in the ... reality, existence ... of sets than in the existence of numbers ? – Mauro ALLEGRANZA Dec 24 '13 at 20:23
  • They got the names they did for a reason. They're not just labels, they're good labels. The question being asked is in part why are they good labels? – Lucas Apr 3 '14 at 4:24
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The introduction of fractional and negative rational numbers may be justified from two points of view. The fractional numbers are necessary for the representation of the subdivision of a unit magnitude into several equal parts, and the negative numbers form a valuable instrument for the measurement of magnitudes which may be counted in opposite directions. This may be taken as the argument of the applied mathematician. On the other hand there is the argument of the pure mathematician, with whom the notion of number, positive and negative, integral and fractional, rests upon a foundation independent of measurable magnitude, and in whose eyes analysis is a scheme which deals with numbers only, and has no concern per se with measurable quantity. It is possible to found mathematical analysis upon the notion of positive integral number. Thereafter the successive definitions of the different kinds of number, of equality and inequality among these numbers, and of the four fundamental operations, may be presented abstractly.(by h.s carslaw)

What numbers we find in nature ? have you find negative numbers ? as name suggests natural numbers are found in nature. say a particular length(say a stick s) is taken as 1 unit of length (e.g 1m) now if there is some other stick(s2) which is equal in length of of two s sticks we say its length is 2 units.similarly length can be of fractional units of s. numbers are labels for representing a particular length. same idea can be extended for all measurable quantities. for -ve numbers consider expression
(a-b)*(c-d)=ac-bc-ad?bd

if 'a' is length > 'b' length and 'c' length > 'd' length then product should be +ve try putting values in the expression you will find the expression holds good if '?'='+' make a square of length a and breadth 'c' then another of length 'b' and 'd' by superimposing 'b' on 'a' and 'd' on 'c' now consider every product in expressing as an corresponding area in diagram . you will recognize soon that '?' should be replaced by '+' or you can create a rule that distributive law holds good if we consider two -ve numbers having a property such (-b*-d) = (+b*d) imagine the importance of distributive law it makes a formula like (a-b)^2=a^2 - b^2 + 2ab. this formula gives us a shortcut to perform calculations which has become possible only if we have -ve numbers of such properties(multiply two -ve number means a +ve product of their magnitude). surely if we don't define -ve numbers we will have lengthy calculation always.

complex no's:

A*sin(wt)=RE[e^{jwt}] this concept is been used a lot of times to reduce calculations like in network analysis which involves impedances.

you should read : Beginning Algebra for College Students Second Edition by Lloyd L. Lowenstein (Author)

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Do numbers exist outside of our heads? No.

Is what exists inside our heads real? Yes.

Do numbers exist? Yes.

If knowing something is real is the definition of what is real, then maybe numbers are as real as anything in the universe.

I have a pet hamster, I love the hamster. Is the hamster real? My experience of the hamster is real, but the hamster may be imagined, such is the nature of dreams that they seem to be real. Such is the nature of numbers that they be nothing but our most fervently dreamed dreams.

But what matters more to the universe, a dream or a rock? Upon this rock we have built our dreams. And without our dreams and the dreams of all things there would be nothing here.

And yet, how is it that I have 2 eyes, and 10 toes? Is it because nature can count? Or is it incidental? What is a toe but a tiny misshapen toe attached to a larger toe? Incidental fleshy appointments adorning a larger fleshy appendage so named and numbered by the coincidence of thought observing its own fleshy body.

Who are you with your fingers and your eyes reading this, and why do you read sir or madam, is it curiosity, fear, love, or something else that drives you today?

Why did you think about what a number was and come here to read about it?

Because, somehow, you want to know if YOU are real. Perhaps you believe you are a number. Perhaps you need something, anything at all to latch onto today, to give you a place to rest your weary mind travelling this vast expanse of possibility.

So many possibilities!

It makes me wonder, what is real. And the realest things we can think of are the things we can trust the most. I think therefore I am, irrefutable. But who are you? I knoweth not who I am, therefore do "I" think? I can't be sure, for it may be another that thinks for me, perhaps I just watch them thinking. And yet I know the number 1. Yes, and if I take 1 of one thing, and another of the same, I'll have 2 of these things. And this I can trust forever and ever... But I started to wonder, is adding things real? Is there really ever 2 of something? When I look do I see with my own 2 eyes 2 different images? No, I see one image, my 2 eyes function as 1. What do I SEE? I see 1 image, therefore, I have one eye in my mind.

So what is a number anyway? Is it a perceptual construct? Is it a definition?

It is a belief. Just like all things, we believe, I believe. I BELIEVE. YOU are I. I BELIEVE IN YOU AND ME. I believe in US. I believe... in numbers.

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I am just adding to the excellent answer given by @Niel de Beaudrap. He questioned the "real versus man-made" dichotomy people overuse. The purpose of this answer is to show some other aspects of the question not addressed already.

  • Are numbers found in nature? (I suppose that is what he meant by real)
  • If not, how can we apply them for real things?

And two minor questions

  • How are imaginary numbers any more imaginary than real numbers?
  • Why can't complex numbers be given a definite order?

Are numbers found in nature?

No. Numbers are not found in nature. You can find 'two apples' in nature but not 'two'. Again it is interesting to note what we mean by saying 'two apples'. Do we mean two objects that are identical? Then we cannot talk about two apples because no apple is like another. So, we are talking about two objects that are similar. "How similar" is the next question. Obviously we want to avoid counting an orange as an apple. But we want to count it when we count fruits. Also we may not count an apple when we are counting "small apples". So obviously, counting is artificial. But so are many other things we take for granted in life. And clearly it is not just real numbers or complex numbers; even counting numbers are artificial. We accept counting numbers as kind of real and question only more artificial ones like real numbers because we are accustomed to counting numbers.

Still, the notions of counting numbers, fractions and amount are highly useful for our purposes today as explained by @Niel de Beaudrap. So, numbers are not found in nature. Numbers help us capture the idea of patterns we find in nature. Note that what we find in nature need not be what is there in nature. It is indeed real to us because our world is what we feel.

If not, how can we apply them for real things?

Well, that's the tricky part. Numbers are tools in mathematics. Branches of science like mathematics and logic are not about the real things; they are not meant to be. They are indeed about the abstract. This is both their power and weakness.

If you give them some rules of a world which may or may not exist, they will tell you a lot of other things about that world. So, if you give them rules (any rules), they will tell you many consequences of those rules. That is their power. This is why they are applicable almost everywhere. And they will tell you only consequences of those rules, the personal beliefs of the oracle has no place there. This is why they emphasize on rigor.

But if you are interested in a world whose rules are unknown to you, there they are helpless. This is exactly true of our physical world as we know it. Physics is interested in the rules of our world but maths cannot provide them. (In contrast theoretical physics and maths are close friends). Therefore you need a bridge between them to make a link. This is a gap only philosophy can fill. And philosophical tools like models are the usual way to go.

Minor questions

How are imaginary numbers any more imaginary than real numbers? Well, imaginary numbers are not an ounce more imaginary than real numbers. In a lecture on complex numbers, the professor asked students to raise hands if they think imaginary numbers are imaginary and real numbers are real. Around thirteen students raised hands. Then he said this, "okay, we can discuss on it. Half of you come to the stage".

Why can't complex numbers be given a definite order? By order, they don't mean a general thing; They are talking about a specific concept called total order. Saying complex numbers cannot be ordered means that whatever ordering you come up with, it will fall short of at least one of the conditions for total order compatible with the usual field operations of addition and multiplication. You can find more details from this question in stackexchange and this page from cut-the-knot. In fact the set {0,1,-1,i,-i} of complex numbers themselves will make problem when we try to give a total order that goes with the usual field operations. I shall give details if you are interested (not hard, but i think it will not be of any philosophical significance to you).

  • The set {0,1,-1,i,-i} is totally ordered just as you wrote it, left to right. There's no order on the complex numbers compatible with its algebraic structure. But there are plenty of total orders on the complex numbers. Lexicographic order on a + bi is one such. – user4894 Mar 24 '18 at 23:31
  • Edited. Thank you @user4894. I was trying to keep the details minimum. – Cyriac Antony Mar 25 '18 at 4:35
  • The definitions for (total) ordering and ordered field can be found in page 246 in Stephen Abbot's book "Understanding Analysis" – Cyriac Antony Mar 27 '18 at 4:39
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We have named them "numbers," but in reality "numbers" is just a human made label for naturally occurring rules and principles. However, wether we call them "numbers," "counts," or whatever other arbitrary name, they would continue to play a key role in the manifestation of reality regardless of our knowledge of them.

If an alien race were to contact us, numbers and mathematical calculations (in some shape or form) would be something we'd have in common. Different ancient civilizations had different numeral systems, but they were "numbers" nonetheless." Even nowadays one can see the evident difference between Chinese numerals (零,一,二,三,四,五,六,七,八, 九) and Arabic numerals (0-1-2-3-4-5-6-7-8-9); despite the difference in symbols, the concept behind them is the same.

The label "numbers" is the attempt to describe the "code of the universe." So, roughly speaking, I'd say yes, numbers do exist.

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If it's all right with you, I'd like to focus on geometry rather than numbers. I feel the same about both areas, but geometry fits a little more nicely with my example.


Consider the statement:

The angles of any triangle sum to 180 degrees.

If you're reasonably familiar with basic geometry, this will appear obviously true.

What about this statement?

James Kirk is captain of the USS Enterprise.

We could claim that it's false, I suppose, but if we are attending a Star Trek convention, that just isn't very polite. But it gets worse. If we claim the above statement is false, we are asserting that:

James Kirk is not captain of the USS Enterprise.

And that still suggests there is both a Kirk and a USS Enterprise, in addition to annoying the Trek fans. There are more complicated ways in which we may interpret the negation operator, but this is not a trivial problem.

Suppose we accept that Kirk is captain, to placate the fans. But then one of them comes to us and says:

I'm a fan of Star Trek: The Next Generation, and I think your Kirk statement is false. The captain of the Enterprise is Picard, not Kirk.

Then, while we're puzzling that over, a mathematician comes up to us and says:

I'm a fan of non-Euclidean geometry. I think your triangle statement is false.


Mathematical statements are true within the context of their axioms. Statements about fiction are true within the context of their canonical sources. If you choose different axioms, or different canonical sources, you get different truths (if the Kirk/Picard example is too subtle, compare and contrast Dracula with Twilight). While mathematics is more rigorous and in most cases more directly useful than fiction, both are forms of art.

Like many arts, both math and fiction aspire to both truth and beauty. But these are aesthetic qualities, not objective realities. Mathematics is "true" when you find a real-world situation that it accurately describes, and apply it correctly. Fiction is "true" when you find that it resonates with your life experiences and goals, and try to live by its teachings. These truths cannot exist in isolation; they depend on the observer to actualize them.

So, to answer your question, numbers, or triangles, are just as "real" as the application you have found for them. But if you're just doing math because you think it's beautiful, then you don't need to care whether it's "real." Maybe someone else will find an application some day, as happened with number theory and cryptography. Maybe not. Either way, worrying about it would be missing the point. You're not doing this for truth. You're doing it for beauty.

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Leopold Kronecker stated that the non-negative integers where made by God. Anything else is "crafted" by humans. Following this idea we know for sure that the non-negative integers are real. Now, the statement "Numbers are real." is equivalent to "Numbers exist." Existence can be proven by writing down one distinct element satisfying the given property. Using that non-negative integers exist, and applying the premise that non-negative integers are numbers, we conclude "Numbers are real."

Edit: What I actually wanted to point out is that the question really depends on how numbers are understood.

On the other hand I would like to to strike a blow for Kroneckers point. In more general terms he described a natural tencency of human beeings to count things. This isn't totally unreasonable. Consider that there were found bones with counting marks that are approximately 30000 years old (I hope you won't blame me if I don't give a bibliographical verification) - long before people thaught about axioms for constructing natural numbers.

  • Argument from authority? – Niel de Beaudrap Oct 31 '13 at 20:59
  • @NieldeBeaudrap, I don't argue with an inductive argument. Isn't the opposite a requirement for the Argument from authority? – Ruben Nov 1 '13 at 0:04
  • "Leopold Kronecker stated that the non-negative integers where made by God" [emphasis mine]. – Niel de Beaudrap Nov 1 '13 at 1:36
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    The fact that humans have used an idea without axiomatisation does not mean that it "exists" independently of humans. Is magic real? Is luck real? – Niel de Beaudrap Nov 1 '13 at 19:48
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    I think that you are allowing yourself to think of the word "use" differently for 'magic' and for 'numbers', but never mind. – Niel de Beaudrap Nov 1 '13 at 21:06
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Numbers are symbols only. They describe things just like words and language does. Numbers are the symbols we use to denote an amount of something not the things themselves. When they are used to communicate an idea then they become a language. The numbers themselves are constructs that we create as tools to work with. To perform tasks with and solve problems with. We can use them also to convey abstract ideas as well. Thus the problem of negative and complex numbers. These numbers are merely ideas using the number symbols to convey information nothing more. They exist only as ideas in our minds. We all simply agree on what to call these ideas and the properties they have. So no they do not exist any more than red or sweet or happy exist as real things. They are descriptors only.

  • Hi, Welcome to Philosophy Stack Exchange! Can you provide more justification for your stance? It seems that after your initial claim, the rest of the statements are just restatements of your initial claim ("Numbers are symbols only"). – James Kingsbery Mar 15 '15 at 1:04

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