Mathematics. What is it anyway?

[LN400]

I'm a curious fellow. Another topic here on the forums or rather a detour in this topic (the metric vs imperial topic, to be exact) stirred up that curiosity. What is mathematics anyway? So, I began looking around.

First stop was ye ol' Wikipedia. Now Wikipedia can be a great source of information but it can be a wild ride sometimes (too often, in fact).

Here is what is driving me up the walls (from Wikipedia's article on mathematical structure):

"In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that..:"

So, to understand structure, I should know what a set is but wait no, a set apparently is only part of something bigger, a type so I should look that up first, or should I look at mathematical objects first, to understand better what a type is?

So what I did was, as soon as I came across anything to the likes of "based on [new concept here]", or "a sub-set of [new concept here]" etc, I followed those links hoping to get to an even more fundamental er, fundament that would explain a certain idea or term. Soon I came across the article on mathematical objects

https://en.wikipedia.org/wiki/Mathematical_object

Now we are rushing towards the realm of philosophy. Still I feel there are even more fundamental terms I should understand before really understanding mathematical objects.

Looking around, it is true that often you will find say an explaination of A that requires a prior explaination of B, which itself refers to A for prior explaination, or to C which in turn refers back to A.

Circular explainations are tricky. Most of the time, it seems, they ultimately fail to explain but one can not rule out that at the very base, there are in fact some A that can only exist if B exists and vice versa, the two are not the same but one can not exist without the other, and together they do form a unit (in lack of a better word) of foundation which all other ideas rest upon.

If anyone here have ever dived into the depth of mathematics, here's a question:

Where would one want to start to learn about mathematics in it's purest form if one wants to start at the very start, where the foundations have no further foundations underneath?

Edited February 24, 2016 by LN400

[RizzoTheRat]

Maths is the foundation of everything else, so it's a good place to start.  Some of it may sound like philosophy but the point of maths is that, unlike in science, we can actually prove stuff.

 

I'm an engineer not a mathematician, but I'd start with number theory if I were you, as it's kind of the foundation of most other maths.

How advanced is your maths background?  If you know a bit of maths and some programming skills then take a look at Project Oiler.  It's about writing bits of code to solve mathematical problems, but once you get past about the first 10 or so you really need to start delving in to the maths to be able to solve them as brute force computing approaches would take ages  https://projecteuler.net/

[LN400]

Thanks for the post. I am not as much interested in solving particular arithmetic problems (altought I am studying engineering and will need just that) but rather interested in the foundations of mathematics itself.

I did come across a (to me rather obscure) article

https://en.wikipedia.org/wiki/Foundations_of_mathematics

From a very hasty read-through it seems this is one heavy topic, which is still investigated and debated. Still, it appears that somewhere in this field of investigation, are some of the answers.

Number theory is indeed a topic I find fascinating (I wish my skills would match my interest) but it appears that even that is a sub branch of a much wider and deeper field.

As for truths in mathematics, that itself is perhaps the main reason I started to look at maths beyond arithmetics. Axioms, and truths, as I learned later, are not carved in stone but rather chosen. An interesting video here:

 

[p1t1o]

I can't be having with reading walls of text about the philosophy of science at this time in the morning so here's a stream-of-thought just off the top of my head (so could be complete rubbish) -

It seems to me that mathematics is an unavoidable part of being a thinking being in a non-homogenous universe. If any part of the universe is different from any other part, you need maths in order to understand, anything. Where am I? I am here. I know I am here and not there, because there is different to here. There are, therefore, at least 2 different places. Woops I invented maths.

To not need maths would be to be in a universe that is completely homogenous, every part is the same as every other part (see heat death of the universe), in this case, it is hard to imagine the concept of a thinking entity existing to need maths. And they would only need the number 1, and there would be no operations.

So, if you have at least, A) a non-homogenous universe, and B) a thinking being in it, maths (or an equivalent concept) MUST arise.

[kerbiloid]

Mathematics is a theoretical level of the magic. It describes the source code of the Universe.

[LN400]

p1t1o: I believe you are right there. To borrow from someone else: When we or any other capable animal, realized that 2 apples and 2 elephants have something in common, namely the two-ness, then we had a foundation to explore what that could signify. The ability to see that 2 apples and 2 elephants have something in common that 2 apples and 3 apples do not have in common, is necessary to form ideas that make up mathematics. So I do believe that is where it started a long time ago.

The meaning of mathematics has changed over the millennia, and explosively so over the last couple of centuries. We got non-euclidean geometry for one thing, even the idea of zero had to fight for acceptance, not to mention complex numbers. Set theory, all sorts of new theories and fields have popped up.

So where are we today and where would one want to start learning about it all? Those are some of the questions I have. But ultimately it comes down to what is underneath it all, that is the base for understanding things like sets, fields, group theory, all which are pretty high level terms.

Edited February 24, 2016 by LN400

[RizzoTheRat]

Not got sound at work but that wolfram vid looks interesting, will have a watch tonight, thanks.

 

Not specifically looking at the background of maths, but if you want some interesting maths videos take a look at Numberphile on YouTube, it's a bunch of mathematicians doing short videos about interesting stuff. http://www.numberphile.com/team/index.html

[p1t1o]

10 minutes ago, LN400 said:

p1t1o: I believe you are right there. To borrow from someone else: When we or any other capable animal, realized that 2 apples and 2 elephants have something in common, namely the two-ness, then we had a foundation to explore what that could signify. The ability to see that 2 apples and 2 elephants have something in common that 2 apples and 3 apples do not have in common, is necessary to form ideas that make up mathematics. So I do believe that is where it started a long time ago.

Exactly, and even more fundamentally, to even recognise the concept that an apple is not an elephant, or even that an apple is different to its immediate surrounding and that the apple is a discrete object, one must be able to separate things into groups: "things that are apples" & "things that are not apples", just by doing this you have created a binary number system, which you can use to define/understand the universe around you.

Without being able to do this, to tell the two groups apart, literally everything will be the same, your entire field of vision/conception will be a field of grey (if grey was even a thing, since now you can't tell "grey" from "apples"). Even if all you could percieve was black and white, and the universe consisted of two totally homogenous halves, one black, one white, and this is literally all that can possibly be understood about the universe, boom, just by observing the dichotomy you have created another binary number system, or, you need to in order to observe them... In fact, without being able to do this [concieve of "binary"] it is hard to imagine being able to determine anything in said universe. Mathematics is a part of being able to even think at all, I think it might be part of the definition of sentience, or possibly not even a seperable part of it.

Woah.

 

 

[LN400]

Rizzo: Great link. I am familiar with Numberphile. One of my go-to channels for a bit of nerding out! Heaps of interesting stuff there for anyone even remotely interested in mathematics.

 

[K^2]

My background is in particle physics. So naturally, I've studied group theory. I also understand fundamentals of General Relativity, so I have basic grasp of differential geometry. I've taken graduate courses in topology and analysis, because these help. Things like calculus of variation, basics of modern algebra, and functions of complex variable go without saying.

I happen to have a few friends who have Ph.D.s in various fields of pure mathematics. I can say with absolute certainty that I know ****-all about math.

Seriously, unless you study mathematics professionally, don't even try to comprehend the depths. it's a super useful field all over the place. But only a mathematician can say what mathematics really is, and only another mathematician will actually understand it.

[RizzoTheRat]

4 minutes ago, LN400 said:

Rizzo: Great link. I am familiar with Numberphile. One of my go-to channels for a bit of nerding out! Heaps of interesting stuff there for anyone even remotely interested in mathematics.

 

In that case also worth watching some of Matt Parkers Stand Up Maths and Festival of the Spoken Nerd stuff, and then getting yourself along to a Maths Jam if they have them in your area (huge level of nerdiness, luckily I married a maths teacher so get to do things like that without complaint).  I've met Matt and James Grime a few times, nice guys.

Edited February 24, 2016 by RizzoTheRat

[LN400]

K^2: I hope I have enough wits with me to know where my wits fail me  but one can always try to seek to understand. Even if it takes a lifetime it would be worth the time, at least to me, to understand only fractions of it all.

EDIT: p1t1o: It is interesting you mention binary as 2 (true/false, is/is not) is the lower limit to how many symbols/values are needed to convey information. It would be very hard to imagine any being capable of awareness of its surroundings or even itself without the ability to have some sense of difference between at least 2 non-identical states. As it is, we clearly do have that ability. On a side-note: There are people who traditionally have words for 1, 2 and 3 or perhaps up to 5. Any number greater than that would be "many". How would they build a world of mathematics if they ever did? I can't say it's impossible, only I have a very hard time understanding how it would be done.

Even what numbers are, is still debated and has been for at least 2500 years. To the Greek, apparently, numbers were not to mathematics what they are today. Shapes mattered, geometry was all there was, they didn't think of 1 as a number, or 2 for that matter. It didn't concern them. That was very different from the Indian, the Babylonians, the Egyptians, the Persians and the Arabs who were concerned with arithemtics rather than "shapes for the sake of shapes".

Perhaps the focus on numbers is a bit of a red herring. It seems to me that there is a world of thought there that exists even without numbers, but one which numbers live in to have any meaning.

Edited February 24, 2016 by LN400

[PB666]

2 hours ago, LN400 said:

I'm a curious fellow. Another topic here on the forums or rather a detour in this topic (the metric vs imperial topic, to be exact) stirred up that curiosity. What is mathematics anyway? So, I began looking around.

First stop was ye ol' Wikipedia. Now Wikipedia can be a great source of information but it can be a wild ride sometimes (too often, in fact).

Here is what is driving me up the walls (from Wikipedia's article on mathematical structure):

"In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that..:"

So, to understand structure, I should know what a set is but wait no, a set apparently is only part of something bigger, a type so I should look that up first, or should I look at mathematical objects first, to understand better what a type is?

So what I did was, as soon as I came across anything to the likes of "based on [new concept here]", or "a sub-set of [new concept here]" etc, I followed those links hoping to get to an even more fundamental er, fundament that would explain a certain idea or term. Soon I came across the article on mathematical objects

https://en.wikipedia.org/wiki/Mathematical_object

Now we are rushing towards the realm of philosophy. Still I feel there are even more fundamental terms I should understand before really understanding mathematical objects.

Looking around, it is true that often you will find say an explaination of A that requires a prior explaination of B, which itself refers to A for prior explaination, or to C which in turn refers back to A.

Circular explainations are tricky. Most of the time, it seems, they ultimately fail to explain but one can not rule out that at the very base, there are in fact some A that can only exist if B exists and vice versa, the two are not the same but one can not exist without the other, and together they do form a unit (in lack of a better word) of foundation which all other ideas rest upon.

If anyone here have ever dived into the depth of mathematics, here's a question:

Where would one want to start to learn about mathematics in it's purest form if one wants to start at the very start, where the foundations have no further foundations underneath?

I think that to get to the bottom of the question you need to look at the natural system, and when you do that our numbers and numbering system look meaningless.

But then once you get there there is a certain elegance

for instance e = m, f = m1 * m2 / r, and alot of complicated physics looks simple and you don't see constants all over the place

So what is a constant, a constant is simply a correction factor between the quantum 'reality' and our world view, which happens to be interpreted primarily with the metric system.

In nature there are symmetry and axis. Our 3D view of nature, reading the Feynman lectures, its not so important, from a space-time perspective, alot of the physics is about the perspective of a point, and its often on a second point, more or less binary.

It gets worse because when you get to the quantum level you start dealing with a statistical world of probabilistic potentials.

We are describing relationships, some, like the mass energy equivalence can be greatly simplified, others cannot be simplified.

You ask yourself the basic question, we have been studying numbers for 10000 years, rather intensely for 4000 years, but it wasn't until about 400 years ago that the first of these constants, G, was inferred and not until Boltzmann and Planck that two more were discovered. These constants are not rolling off the tongue or apparent, you have to dig deeply to see them, we are in the process of discovering the fundamentals of nature which have relationships that can be described by math (but see above, just by change foundation I could make the mathematics change) . . . . but I should point out gravity may not be constant, that is to say the degree that space-time warps space may change depending on some metric for the local or total universe.

[LN400]

PB666: Thanks for a thorough post. Something that Wolfram (in that video above) talked about which caught my attention was (if I understood him at all correctly) was, he argued that the reason we see mathematics as the language of the Universe is because we happened to choose a particular mathematics in a space of all possible mathematics (I have no idea what such space would be like, but I just started out!) which can describe what it can describe (circular), and we tend to think that is all there is. But as he points out, there might very well be other, radically different mathematics that can very well describe what we may not be able to describe (mathematically), while failing to describe what our mathematics can. This is a peculiar view, to say the least.

 

EDIT: Somewhat on this note, a discussion over Gödel's theorems which go right to the truths (or rather how you can't really tell if you have the truth) in any set of axiom.

 

Edited February 24, 2016 by LN400

[WinkAllKerb'']

2 hours ago, RizzoTheRat said:

Maths is ...

 

... a tool , like most tool it's imperfect amongst time and evolve and can't solve everithings @ lambda dt ; )

"Welcome aboard the math plane, fasten your sit belt and enjoy the flight"

Edited February 24, 2016 by WinkAllKerb''

[LN400]

Maths is certainly a tool, and a powerful tool to boot. Whether or not it is merely a tool, or if there is some kind of deeper truth underneath is something that is debated. Or, is our mathematics the only mathematics? It seems to me most would say yes. Our maths is the only maths. Still, that is up for debate as well. What other tools are in the tool box named The Space of All Mathematics?

[Bill Phil]

1+1=2 is probably the purest statement there is. One operation, and it's the base operation of the others. Only single digit numbers. 

If you want to get proficient you should get a degree in mathematics. Some of this stuff is gonna require higher level math.

[LN400]

Bill Phil: ...or philosophy, or preferably both mathematics and philosophy.

One thing is arithemtics under the axioms we have. Another thing is the greater world of mathematics as a whole. Take axioms, they are supposedly all powerful yet Gödel said that no set of axioms could ever prove itself and its items entirely. Take the problem of undecidability. P vs NP problems. These problems affect mathematics at its core.

Then add to that the debate whether they really are about mathematics at all or rather our incapacity to fully comprehend one, final mathematical world.

[WinkAllKerb'']

this is where you need an integer with delta t:

Integer:(being "n",lifespan).dt ; )

 

[kerbiloid]

Then probably the purest one is an increment operation "increment by one".

Because 0++ (or ++0) results 1 from scratch, and it's an unary operation.

And the Archimedean property is the root of all things.

[WinkAllKerb'']

bin array k ally ; ) yup ; ) does it solve anithing for a mathematician: nope ; ) but from a philo  pov that's one more step ; )

so on the main diff with philo and math is there relation with time and it's uncertaintity : ) and the ability to solve evertithing right now frustration ; )

Edited February 24, 2016 by WinkAllKerb''

[sevenperforce]

21 minutes ago, Bill Phil said:

1+1=2 is probably the purest statement there is. One operation, and it's the base operation of the others. Only single digit numbers. 

Blasphemy! 1 + 1 = 10 is far purer!

[WinkAllKerb'']

hexa are like black hole ; ) matteer condensed 10+1=A binary is like the cloud around the black hole ; )

[K^2]

3 minutes ago, sevenperforce said:

Blasphemy! 1 + 1 = 10 is far purer!

I prefer 1 + 1 = 0. Because it's a finite simple group of order two.

[LN400]

1+1=2 was mentioned. That leads to a host of questions as well. What are the properties of 1-ness, what are the properties of 2-ness and are those properties valid in all mathematical worlds? It may sound ridiculous but these are just 2 (yeah I know) questions that have been looked at real seriously by number theorists, by philosophers. The properties of numbers is just one field of study in mathematics and one that has led to a number of real life applications, like say the ciphers that protect your online bank transactions.

Edited February 24, 2016 by LN400