Do You Think Mathematics Is Natural Or Man-Made?

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!

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

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Originally Posted by ReachingForTheDream

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.

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Originally Posted by ReachingForTheDream

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.

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Originally Posted by Supernova

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

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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.

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...

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?

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

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Originally Posted by stonedape

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

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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.

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.

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Originally Posted by ninja9578

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.

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exactly my feeling. In reality math is just our representation (approximation) of natural phenomenon. as such...it is both hehe.

My thoughts...


Math is a symbolic "language" created by mankind in order to measure and therefore control the world around him. The natural world does not contain lines ,points or planes... these are purely imaginary ideals, but can be arranged to describe a perfect version of things in nature... IE a wall is not a plane, but a surface of a wall is like a very imperfect plane, thought it's not perfectly flat and has texture etc.

I think there's a fine distinction between doing math and "adding things together" in the natural world. Yes, you can pick up a rock and place it with another group of rocks, but unless you understand the purely abstract language of math, you're not adding anything, you're just moving rocks around. To make it math requires a certain intelligent understanding of what the symbols (rocks, numbers, whatever) represent.

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Originally Posted by ninja9578

It's also the one true science.

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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.

One large tautology?
y/n?

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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.

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But 2 + 2 = 4 isn't mathematics. It's a tautology.

Mathematics is about inductive systems based on axioms we have no right to assume are true other than human experience. Mathematics can be made to contradict itself in this way. Mathematics is also limited. There are true mathematical statements which can't be proved with mathematics.

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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?

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It has been empirically proven that Euclidian geometry is incorrect. It's often an extremely good approximation but curved space is better. There's really no ambiguity, it's science.

Like I said, you can deduce the value of pi based on Euclidian axioms (and perhaps some others, I'll use some from analysis). I suppose you could use the Newton-Raphson method. This will give us an approximation for pi.

f(x) = x - (sinx - 1)/cosx

If you repeatedly perform this function on a number around 1, your calculator will show you a value which gets closer and closer to half of 3.1415926535... (which you then double).

There are many other ways of finding pi but the point is they are all somehow dependent on Euclid's axioms. I'm interested in how you think we work out what pi is in mathematics. To be honest the whole discussion's a bit dodgy because actually all we ever do is give approximations to pi, talk of 'calculating' pi is not exactly right.

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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.

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That's not true. Your value of pi will be very different in extreme situations, but it will still not be exactly the value of Euclidian pi pretty much anywhere in the universe. The difference will be miniscule but it is there.

pi in base-10 does not have a different value in base-11, it has different digits. These are totally distinct concepts. The digits are just how we represent pi and are essentially a human invention. pi itself has a fixed value and is an incontrovertible consequence of Euclid's axioms.

It's true that aliens will likely have discovered the same theorems. This is because they will have observed that the same axioms hold in their part of the universe.

That still doesn't answer my question. Why didn't they make pi = 1?

We've had this conversation many times, but I will say it again. Euclidian geometry is taught at every accredited high school in the United States and maybe even the Western Hemisphere, and it is 100% logical. Its concepts apply to calculus and trigonometry and are used by all architects and engineers. Planes do exist in the real universe, even though they are not made of matter.

Pi is derived from the ratio of any circle's circumference to its diameter. Like the Heisenberg Principle and other insane ideas of quantum physics and other modern WTF, the rejection of Euclidian geometry is new age hocus pocus that is not going to stand the test of time.

Since you reject Euclidian geometry, I will also make my point with another question. Why didn't they just make the principle square roots of whole numbers equal whole numbers in every case?

I answered your question exactly. 1 is not the result of the iterative formula I gave; a formula which is a direct consequence of Euclidian axioms.

Of course schools teach it. Of course architects use it. On Earth it is more accurate than you could ever measure in all but the most exotic situations.

If you reject the curvature of spacetime you have rejected the scientific method.

http://en.wikipedia.org/wiki/Tests_o...ral_relativity

With regards to the test of time: the theory is a century old.

Your next question is not related to Euclid as much as arithmetic. Arithmetic is based on Peano's axioms. These axioms can be used to prove that square roots of non-squares are not only not whole but not fractions either. The proof is elementary:

http://www.homeschoolmath.net/teachi...irrational.php

You didn't answer my question by the way.

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Originally Posted by 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?

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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.

No that would be you making stuff up.

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Originally Posted by Xei

No that would be you making stuff up.

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If you don't lighten up, you are going to kill yourself early with all the stress of being so serious all the time. Or, is that just your internet facade?

http://serenitythruhaiku.files.wordp...son_cooper.jpg

Math is simply a tool which helps us deal with reality abstractly, and like all tools it's man-made.

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Originally Posted by Universal Mind

Why didn't they just make pi = 1?

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Because in a mathematical system where pi = one it'd be very hard to tell someone how many apples they're holding.

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Originally Posted by Xei

I answered your question exactly. 1 is not the result of the iterative formula I gave; a formula which is a direct consequence of Euclidian axioms.

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... which can't just be changed. They have to fit reality.

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Originally Posted by Xei

Of course schools teach it. Of course architects use it. On Earth it is more accurate than you could ever measure in all but the most exotic situations.

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It is not taught as a subject of approximations.

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Originally Posted by Xei

If you reject the curvature of spacetime you have rejected the scientific method.

http://en.wikipedia.org/wiki/Tests_o...ral_relativity

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I don't reject it. I am just saying planes exist. They are not matter, and they are not space. They are just in space.

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Originally Posted by Xei

With regards to the test of time: the theory is a century old.

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Humanity is 200,000 years old, but still very young.

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Originally Posted by Xei

Your next question is not related to Euclid as much as arithmetic. Arithmetic is based on Peano's axioms. These axioms can be used to prove that square roots of non-squares are not only not whole but not fractions either. The proof is elementary:

http://www.homeschoolmath.net/teachi...irrational.php

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As with pi, why can't the rules it's based on be changed? It's because they have to be in line with reality that is not man made.

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Originally Posted by Xei

You didn't answer my question by the way.

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Which one? The one about pi? I did answer it. I said it's the ratio of a circle's circumference to its diameter.

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It is not taught as a subject of approximations.

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General relativity is correct. K? I don't see any point in this line of discussion. There's empirical confirmation, end of. I don't care who teaches what in what manner.

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I don't reject it. I am just saying planes exist. They are not matter, and they are not space. They are just in space.

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I don't see how that makes sense. Clearly these planes only exist in your imagination.

How about spheres? Do they exist? Is there a continuum of spheres of every concievable radius permeating every point in the universe? What are the empirical consequences of these spheres? They don't do anything.

What about other mathematical objects? Is every single value on the real number line somehow squashed into every point in space? What about the complex plane? What about all of the n-tuples? I can invent any kind of invisible shape or object I want and say it 'exists in space'. Why are planes special?

If that comes across as sarcastic it wasn't supposed to be, I just don't understand what your position is.

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As with pi, why can't the rules it's based on be changed? It's because they have to be in line with reality that is not man made.

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Well, 'square root' is something only with meaning in arithmetic. So you'd need to have already assumed the axioms for your question to make any sense.

The rules pi is based on can be changed. Specifically the parallel postulate which when assumed gives you Euclidian geometry and when not assumed gives you other geometries.

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Which one? The one about pi? I did answer it. I said it's the ratio of a circle's circumference to its diameter.

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No you didn't. That's the definition of pi.

The question is how we go from there to obtaining an approximation for pi. If you didn't know what the value for pi is, how do you think you would obtain the digits 3.14159265358979323... ? That's my question.

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Originally Posted by Universal Mind

bla, bla

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You totally ignored my argument. That's not very nice -.-

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Originally Posted by Xei

http://serenitythruhaiku.files.wordp...son_cooper.jpg

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I gotta say Xei, your perpetually severe and humorless attitude saddens me sometimes. Especially in this case, since the particular reason why I ended up posting such a non-contributing joke as my response to this thread is because I've tried many times to argue the point that you are making but have never had the full breadth of knowledge to do it as well as you have been here. I'm not sure if you're A. Cooper picture was a continuance of this attitude or an attempt at humor or what but sometimes I wish you and I didn't disagree on so many things so that you would be a little more receptive towards me when we actually do agree.

I don't mean to come across as humourless and I'm not a humourless person IRL. But I do come here to talk philosophy and learn from others. There's a difference between when I'm posting a purely logical philosophical argument and when I'm actually 'speaking' if you understand me.

I guess in the same way that I'm capable of writing sterile mathematical proofs but in no way am I a sterile person.

I might not be making much sense, I've been drinking and smoking a fair bit.