Do You Think Mathematics Is Natural Or Man-Made?

[ReachingForTheDream]

I was just thinking about natural things and thought of this. Is math man-made or natural? Yes, it is "created" by us. We created our own numbers, letters, symbols. We have our own words to communicate their meanings. But. When I think about it... it still exists in nature. If I have a group of objects, and I take away from them, it becomes lesser. If I add to them, it becomes greater. If I add the same amount, it is doubled, multiplied by 2. If I divide them, they are divided by 2. These are all basics of math. And so on, I think math is nature. Yes, we have our own letters, numbers, and symbols, but to me they are simply tools to manipulate and understand this math to a greater degree. Before languages existed, 2 apples was still 2 apples, even if "2" was not an invented word. Anyways, I gotta run, I'll add more later maybe... and would like to hear your opinions!

[Supernova]

math itself exists naturally, we just created the terms to describe it.

[ninja9578]

math itself exists naturally, we just created the terms to describe it. I couldn't come up with a better answer myself so I won't even try.

[Universal Mind]

If I have a group of objects, and I take away from them, it becomes lesser. If I add to them, it becomes greater. If I add the same amount, it is doubled, multiplied by 2. If I divide them, they are divided by 2. These are all basics of math. And so on, I think math is nature. Yes, we have our own letters, numbers, and symbols, but to me they are simply tools to manipulate and understand this math to a greater degree. Before languages existed, 2 apples was still 2 apples, even if "2" was not an invented word. math itself exists naturally, we just created the terms to describe it. I agree. If people made up math, we could make pi = 1 instead of that insane number. Wouldn't that be a lot easier? This many ** plus this many *** is this many *****. That is a fact that people did not create.

[Xei]

It's a very good question. I don't know. It's possible to argue that it isn't on the grounds that you can have contradictory facts in mathematics. There's a very interesting problem about the different sizes of infinity which is relevant to this. Basically we can consider the set of 'natural numbers' 1, 2, 3, 4, 5, ... as being infinitely big. It's then natural to wonder how big the set of 'real numbers' is, which is basically all the natural numbers and the numbers between them; essentially we're allowing for decimals. Obviously it's infinite, but is it a bigger infinity? It seems like it might be, because intuitively the naturals have this kind of 'blockiness' about them whilst the reals seem 'smooth'. What we do is try and see if it's possible to set up a one-to-one correspondence between the naturals and the reals. This turns out to be impossible. It's easy to prove by contradiction. Below is some hypothetical one-to-one correspondence between the naturals and the reals: 1 <-> 0.3947511039... 2 <-> 4.7461014938... 3 <-> 1.3019840011... ... Now what we do is create a new real number R such that the first digit of R is different from the first digit of the real which corresponds to 1, the second digit is different from the second digit of the real which corresponds to 2, and so on to infinity. We can do this for example by saying 'if digit is 0 change it to a 1; if it's not a 0 change it to a 0'. So here, R = 1.01... Now it's clear that R is not in our list because it differs from the nth natural's real in the nth digit; but R is a real. Therefore it is impossible to 'squeeze' all of the reals onto the naturals. The infinity of the naturals is called Aleph-0 and the infinity of the reals is called Aleph-1. The key question is if there are any sizes of infinity between Aleph-0 and Aleph-1, which is called the continuum hypothesis. It's actually provable that this can't be proved! However, for some proofs, it's essential that you assume it's true, and for other proofs, it's essential you assume it's false. In this way, we end up with some things being true in 'some mathematics' but false in 'other mathematics'. It's hard to see how this can be reconciled with the idea of objective mathematical truth. This whole problem just outlines the larger problem with mathematics, which is that you can't prove everything without some basic assumptions, called 'axioms'. Take for example UM's example of pi. You can't prove what the value of pi is before first making some elementary assumptions which can't actually be proved in themselves, but rather are deemed to be 'obvious' based on human observation. In this case the assumptions are Euclid's axioms, and include things such as 'a straight line can be drawn by connecting two points'. From these it follows incontrovertibly that pi = 3.1415926535... Interestingly Euclid's axioms are actually wrong for our universe; this is based on Einstein's theory of General Relativity. Due to the curvature of space itself, pi does not have to equal 3.1415926535...

[Universal Mind]

I can't say I understand the point about R, but I will look into it more. I also have major doubts about the rejection of Euclidian geometry's application to real space, as you and I have talked about a lot. For now, I will just ask you this... Why didn't they just make pi = 1?

[Xaqaria]

I can't say I understand the point about R, but I will look into it more. I also have major doubts about the rejection of Euclidian geometry's application to real space, as you and I have talked about a lot. For now, I will just ask you this... Why didn't they just make pi = 1? Well for starters, that would mean the circumference of a circle would always equal its diameter. Too many rules would have to change for this to work and it is is a law of nature that mathematicians are inherently lazy.

[Xei]

No that would be you making stuff up.

[StonedApe]

I think it is both natural and man made. It occurs naturally and man makes observations and finds patterns.

[ninja9578]

I think it is both natural and man made. It occurs naturally and man makes observations and finds patterns. Best answer. It's also the one true science. All other forms of science work on theory that may be 99.99999...% correct, but we can never really know for sure. 2+2=4 is fact and always will be. Even in a singularity, where even fundamental physics and chemistry break down, math remains constant.

[Maria92]

Any time you take the circumfrence of a circle and divide by its radius, you get pi. This holds true at every point (save for a few very extreme exceptions) in the universe. It is universal, but also man-made. We have our own terms to describe it. Pi on a base-10 number system is different in value from pi on a base-11 number system, for example. However, when converted to the other number system, they are identical. So yes, math is universal, 2+2 will always equal four at every point in the universe (except for rare exceptions), and even if extraterrestrial lifeforms have developed their own math system, it still renders the same results we get.

[tkdyo]

Best answer. It's also the one true science. All other forms of science work on theory that may be 99.99999...% correct, but we can never really know for sure. 2+2=4 is fact and always will be. Even in a singularity, where even fundamental physics and chemistry break down, math remains constant. exactly my feeling. In reality math is just our representation (approximation) of natural phenomenon. as such...it is both hehe.

[DuB]

It's also the one true science. Math cannot be a "pure" science, because math is not science at all. That's why I named this forum Science & Mathematics. They are distinct entities.

[ReachingForTheDream]

I think it is both natural and man made. It occurs naturally and man makes observations and finds patterns. Yes, man may notice the patterns, however the patterns still exist whether we notice them or not. I'm not gonna get into any of the other arguements, I'm no math genius and have no clue what you guys are talking about

[gaia]

My reflex-reponse upon hearing claims of "absolute truth" : man is NOT the center of the universe. Forced to accept that in the physical sense we cling all the more dearly to the hope that at least the thought-world still revolves around us. That as of a preliminary response. I often find that what is a lot more important in any (semi)philosophical discussion is the intuition being aimed at, in other words prejudice. So better expose it early, which I tried to do above. Here's a question: how will you tell the difference between the two positions, that mathematics is manmade, or that it is universal? Is there anything that all observers can agree upon that is different in these cases? I suspect no. I suspect that the only difference is a lessening of that human, all-too-human attempt to look away from the absurdity of our existence.

[Photolysis]

Responding to the original topic here: I would say mathematics is both natural and man-made in varying quantities. Whilst numbers themselves are an abstract concept to represent the quantity of an item or a value, in some cases these values are natural, such as Pi. Some of the numbers being manipulated are derived from human constructs, some are from natural phenomenon*. *okay technically you could argue that even man-made items are natural, but for the purposes of this discussion we are treating the two as separate.

[Indeed]

Maths are like colour. They've been there, we just gave them names.

[Xaqaria]

When I think about it... it still exists in nature. If I have a group of objects, and I take away from them, it becomes lesser. If I add to them, it becomes greater. If I add the same amount, it is doubled, multiplied by 2. If I divide them, they are divided by 2. These are all basics of math. The problem is that there is no intrinsic object in nature. We decide what is a discreet thing and what is not. Therefore, in order to count a group of things you must first define what is one thing and what is another thing. Someone else may very well decide that what you count as two... diamonds for instance, are really not two diamonds but xbillion carbon atoms, or what have you. The designation of things to count is in itself a construct created by humans. several people have brought up the example of pi. While no one "decided" what pi is, what we did decide was what a circle is, and what a circumference is and what a radius is. Pi is merely the result of these decisions. If someone were to claim that a circle is not something that is decided because it is merely all points on a plane that are equidistant from another point, then I can easily say that we decided what points are, and how to define distance. It all comes back to the mental constructs that we use to define our world. No, they are zones of reality, like the boundaries of a football field that is part of a bigger field and is not marked by anything drawn. Do you mean to say that you believe "football field" to be an intrinsic part of reality or is it merely defined by the rules of the game set forth by it's creators? If it is intrinsic, does this mean that Australians live in an alternate universe because their idea of what a football field is has different dimensions (and is shaped like an oval)?

[Xei]

I basically agree with that.

[BrerRabbit]

Unless one is directly perceiving the true nature of reality, one is perceiving a man-made concept. Concepts, ideas, notions and the like are useful because they help us to classify and to discuss our perceptions. In this sense, the notion of a chair, or of a cloud, or of a symphony is no less different than the notion of, say, an integer, or a differential, or of a connected bipartite graph. In fact, we are especially fond of creating concepts to describe concepts. Math is full of this, but this does not make math any more special than, say, the idea that a string of words can be called "a poem." From this perspective, math is entirely man-made. But so is everything else. Cheers, -BR

[Oneiro]

Quote: "2+2=4" Well.. it depends how you conceptualise it. If I had 4 blobs of bluetack, I could prove that 2+2=1 by moulding them all together. If I had 81 blobs of bluetack 75+6=1 would be a possibility etc etc. I demonstrated this to my Maths Professor friend and won a bet with her. It's all in how you think about it. Another one: take an atom (2 protons) and add 1 proton. You still only get one atom. So, in this case, you could also conceptualise that 2 protons+proton = atom, which could be written as 2+1=1 etc etc. There are loads of these examples. Of course, this doesn't disprove that 2+2=4, it just gives another conceptualisation to it. Back to the OP. Having learnt yesterday about a theory of looping time, theoretically someone in the past/future may have already devised Pure Mathematics and used it to create our universe. Well.. it's a theory! )

[DrunkenArse]

You professor friend probably conceded the bet because somebody who is a professor in mathematics should have known better than to enter the bet without agreeing to a definition of 2, 4 and +. Math cannot be a "pure" science, because math is not science at all. That's why I named this forum Science & Mathematics. They are distinct entities. Thank you! I can't begin to tell you how many times I need to try to explain this. One operates by proof. The other operates by empiricism (either falsification or confirmation depending on who you ask). End of story. The problem is that there is no intrinsic object in nature. We decide what is a discreet thing and what is not. Therefore, in order to count a group of things you must first define what is one thing and what is another thing. Someone else may very well decide that what you count as two... diamonds for instance, are really not two diamonds but xbillion carbon atoms, or what have you. The designation of things to count is in itself a construct created by humans. I'm not sure that this has any content. I see where you are going and generally agree but the same rules of counting apply regardless of if we designate the subject as two diamonds or xbillion carbon atoms. several people have brought up the example of pi. While no one "decided" what pi is, what we did decide was what a circle is, and what a circumference is and what a radius is. Pi is merely the result of these decisions. If someone were to claim that a circle is not something that is decided because it is merely all points on a plane that are equidistant from another point, then I can easily say that we decided what points are, and how to define distance. It all comes back to the mental constructs that we use to define our world. This is more on point. The point of math though is that after we agree upon our definitions and axioms, the rest is inescapable. This is why I agree that an extraterrestrial race would end up with the exact same knowledge(see edit) about mathematics as we have. They may have started with a different set of definitions and axioms but the content of their math would be the same. We frequently consider different ways of measuring distance. For example, a circle is only recognizable as such using the euclidean metric, d(x, y) = sqrt((x1 - y1)^2 + (x2 - y2)^2). But what if we use the metric d(x, y) = max(|x1 - y1|, |x2 - y2|)? Then a circle looks like a square. A surprising amount of topology still goes through. It may seem like that metric is contrived but it's a natural metric for measuring function spaces. where if f1 and f2 are bounded functions on some set S then we can define their distance from each other as d(f1, f2) = sup(|f1(x) - f2(x)|, x in S). The former is straight forward specialization of that for the case of functions on {1, 2} which is all points in 2-space really are. (sup is sort of like max but works when there are an infinite amount of elements that might not ever achieve a strict maximum. Think of the image of cos on (0, 1]. Here, there is no maximum but the surpremum is 1. The supremum is the maximum for the case of finite sets. ) The deeper you get into mathematics, the more you see how universal it really is. Any failure of our particular knowledge of mathematics is merely an artifact of that particular set of definitions and axioms not yet having proved themselves useful to us (in a mathematical, not practical sense). EDIT: By "exact same knowledge", I mean that we would not have accepted contradicting claims and our mathematics could be extended to include everything which they've discussed and vice versa.

[Oneiro]

You professor friend probably conceded the bet because somebody who is a professor in mathematics should have known better than to enter the bet without agreeing to a definition of 2, 4 and +. What I actually bet her was that I could prove that "one plus one equals one" (spoken, not written). She probably fell for it because I used such basic units, and there was no money involved. She tells me she has used it to great effect with her colleagues. I've told her to call it "The Oneiro Conundrum". Heh. BTW.. "..a definition of 2, 4 and +." Now, I understand how 2 and 4 can be defined in different ways, but +? Apart from the obvious, what other definition could there be? Does the symbol have more than one possible meaning? (I'm not a mathematician, as you probably can tell).

[DrunkenArse]

What I actually bet her was that I could prove that "one plus one equals one" (spoken, not written). She probably fell for it because I used such basic units, and there was no money involved. She tells me she has used it to great effect with her colleagues. I've told her to call it "The Oneiro Conundrum". Heh. This makes more sense. For example any mathematician would know that 2 + 2 = 1 mod 3 but there's no n so that 1 + 1 = 1 mod n. 1 + 1 = 0 mod 2 and 2 for all greater n. The integers mod n is pretty much the canonical example redefining +. BTW.. "..a definition of 2, 4 and +." Now, I understand how 2 and 4 can be defined in different ways, but +? Apart from the obvious, what other definition could there be? Does the symbol have more than one possible meaning? (I'm not a mathematician, as you probably can tell). Yes, the symbol has a lot of potential meanings and they're all tied to the set of elements that it operates on. The relevant definition here is "Group". A group is a set of elements, G and a definition for + on those elements. We would right the group (G, +). + has to satisfy four axioms and its conventional to reserve the actual symbol + for those operations which satisfy a fifth. (closure) a + b is in G (associative) (a + b) + c = a + (b + c) = a + b + c (identity) There exists e in G such that e + a = a for all a in G (inverse) for all a in G, there is an element B such that a + b = e. If the operation further satisfies a + b = b + a Then we call it an abelian group and write the operation as +, the inverse of a as -a and e as 0. Otherwise, we write the operation using infix notation (that as ab instead of a + b), the inverse of a as a^-1 and e as 1. Your example of adding clay fails to be a group because there are no inverse elements. That is there is no amount of balls of clay that I can add to any given amount of balls of clay to get zero balls of clay. Unless of course you have balls of anti-clay. It is a monoid though which just means that it satisfies all the other axioms except having an inverse. The natural numbers are the canonical example of a monoid. Some examples of groups: The integers, rationals, reals, and complex numbers, all with addition. These would be (Z, +), (Q, +), (R, +) and (C, +) The cartesian product of the integers with themselves (so elements of the form (a, b) where a and b are integers) with addition defined "point wise". So (a, b) + (c, d) = (a + b, c + d) where the inner addition is the regular addition. The set of continuous functions on [0, 1] with addition again defined pointwise. That is (f1 + f2)(x) = f1(x) + f2(x) The set of rotational symmetries of a square. We can rotate it pi/2, pi, 3pi/2 and 2pi=0 radians. a + b is defined as rotating a radians and then rotating it b radians. If I rotate it pi/2 radians and then 3pi/2 radians, I've rotated it by 2pi=0 radians. The set of rotational symmetries of a circle. I can rotate it by any amount of radians betwee 0 and 2pi and the inverse of a rotation a is the rotation 2pi-a

[Oneiro]

Well that's as clear as mud, but thanks for giving it a bash anyway.