Mathematics. What is it anyway?

[p1t1o]

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.

Yeah I was thinking about the definition of "1" before, but it was baking my noodle so hard I couldn't figure out even how to phrase the question properly.

Something like: How do you define "1" without using any terms which also mean "1" or "1-ness" (ie: "single", "unitary" etc...)

"The smallest non-zero integer" - however, this requires knowing what an integer is, which probably includes knowing what "1" is. 

Even the concept of "non-zero" seems problematic here.

 

Perhaps it is so fundamental, that "1" is its own definition. The "quanta" of mathematics, essentially.

It is very hard to imagine a universe existing in which "1" did not exist, or did exist but had a different meaning to our universe.

But then, just because sometihng is "hard to imagine" this does not mean it is impossible.

 

Oh god my head.

Edited February 24, 2016 by p1t1o

[LN400]

12 minutes ago, p1t1o said:

Oh god my head.

I know the feeling  Luckily far greater minds than mine also know the feeling so I'm not too worried about it!

Surprisingly non-trivial, these questions. Bright people haven't even agreed on whether numbers are real outside the human brain or not!

 

EDIT: For anyone interested with 8 hours and 40 minutes (!!!) to kill, one such great mind's thoughts (not the final word in the matter but one I want to learn more about).

 

Edited February 24, 2016 by LN400

[p1t1o]

1 minute ago, LN400 said:

...whether numbers are real outside the human brain or not!

What are you doing to me?!

[YNM]

Mathematics is just a tool, a convention. The reason why 1+1 = 2 have only small impact on what happens in real life. Say, when you have one apple, then you add another one apple, it becomes two apple. But what if I add one apple with one pear ? One melon ? One... I don't know, anything else but apple ? What would it be ? Two apples (certainly not) ? Two anything else ? No way to say that.

That's why it says that math is about sets. The very heart of math comes in counting a particular set - apples, pears, melons, chairs, fingers, money, anything - and it may (and equally may not) be applicable everywhere.

 

[LN400]

13 minutes ago, YNM said:

Mathematics is just a tool, a convention. The reason why 1+1 = 2 have only small impact on what happens in real life. Say, when you have one apple, then you add another one apple, it becomes two apple. But what if I add one apple with one pear ? One melon ? One... I don't know, anything else but apple ? What would it be ? Two apples (certainly not) ? Two anything else ? No way to say that.

That's why it says that math is about sets. The very heart of math comes in counting a particular set - apples, pears, melons, chairs, fingers, money, anything - and it may (and equally may not) be applicable everywhere.

 

YNM: Mathematics is more than equations, arithemtics, algebra and geometry. Forgive my lazyness by just posting this link showing just a few questions that mathematicians/philosophers ponder over:

https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Recurrent_themes

These questions don't have simple solutions in the form of numbers.

Edited February 24, 2016 by LN400

[WinkAllKerb'']

binary is a model per contrast/complementarity ;

contrast do not apply to time ; but complementarity does

the only contrast you can apply to time is your own lifespan complementarity.

Edited February 24, 2016 by WinkAllKerb''

[RainDreamer]

Perhaps math is the language of reality? 

[WinkAllKerb'']

from a derivative integrated point of view division results in multiplicating the exodus of addition and substraction, so it does following most probability and statistic ; )

Edited February 24, 2016 by WinkAllKerb''

[PB666]

Math is the language we would speak if it were not for scarey snakes, interlopers and delicious bananas. Physics would be a second tongue we would speak if it were not for religion, biology would be the language we would speak if we were not enamored by sin.

[tomf]

One approach to define 1 and the other natural number is using set theory  https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

0={}, 1={{}}, 2={{},{{}}}, n={} Union n-1 which I think (but certainly don't understand) is most of what you need to build up the rest of mathematics

[Camacha]

4 hours ago, K^2 said:

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.

While I do understand what you are trying to say, it also sounds an awful lot like a no true Scotsman argument. Religious people and parents tend to use a similar argument against people not in their club.

I am not trying to be offensive, mind you.

[wumpus]

4 hours ago, K^2 said:

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.

I was going to respond that mathematics is what you get out when you pour coffee into a mathematician (the traditional response being "theorems"), but note that it still takes another mathematician to understand the answer.

Note that "super useful" is a bit tricky.  Mathematicians use math to prove theorems.  Pretty much everybody else uses math to calculate results or using certain mathematical properties for similar purposes.  But yes, mathematics to a mathematician is vastly different than math to anyone else.

[YNM]

1 hour ago, LN400 said:

YNM: Mathematics is more than equations, arithemtics, algebra and geometry. Forgive my lazyness by just posting this link showing just a few questions that mathematicians/philosophers ponder over:

https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Recurrent_themes

These questions don't have simple solutions in the form of numbers.

 

I'm no philosopher - what I do is observing. Not even concluding.

Edited February 24, 2016 by YNM

[p1t1o]

On the topic of maths-as-a-fundamental-part-of-reality: has anyone read any of Alastair Reynolds’s Revelation Space series?

In his universe there are machines called “cryo-arithmetic-engines”. Basically, in his universe, somebody discovered a bizarre set of calculations that *just by working out the result* leads to a local violation of the second law of thermodynamics, lowering the local temperature. Run these calculations repeatedly through a computer and you have a machine which literally “sucks up” and disposes of heat.

This, as you can imagine, has great utility for things like super space engines, heavy industry, and especially for space-stealth.

They do have a rather weird failure mode though. A cold computer runs faster. A functioning cryo-arithmetic-engine has complex redundant feedback controls in place. If these are damaged, the machine may get too cold. This makes it run faster, which makes it colder still etc.. in one case a stealthed ship crashed into the sea and within minutes was encased in its own iceberg which showed no signs of the growth slowing.

Its handwavium at its finest, but the idea that a *calculation* might have a physical effect is a fascinating one!

Edited February 24, 2016 by p1t1o
spelling

[WinkAllKerb'']

a fun thing historically and loccally as anecdot:

in french dictionnary math is gendered "feminine plural noun", it's almost never used in it's singular form within sentence and most sentences use the plural form. so if math not dramatical at all it's gramatical on it's own it use.

the fun most mathematician are male according to statistic ; )

 

Edited February 24, 2016 by WinkAllKerb''
localized // approximate

[K^2]

3 hours ago, Camacha said:

While I do understand what you are trying to say, it also sounds an awful lot like a no true Scotsman argument. Religious people and parents tend to use a similar argument against people not in their club.

I am not trying to be offensive, mind you.

The distinction is that you don't have to be a mathematician to reach the conclusion. It is enough to study just a little bit of what math is really about.

Another thing to say about mathematics is that there is an entire field of applied mathematics. These people study how to understand mathematical theorems and turn them into formulas the rest of us can use. I've never heard of applied priests outside of Dungeons and Dragons.

[LN400]

Applied mathemathics and pure mathematics have their equally important places and I would say one would be rather empty without the other. Applied mathematics without pure mathematics is doable but would be less rich while at the same time, pure mathematics without applied mathematics would be a bit like a being incapable of interacting with the suroundings, Would be a dreadfully dull dinner guest that.

Then there are people trained in pure mathematics who came up with mathematical algorithms that had very real and very much wanted applications. Clifford Cooks is one. From a background in number theory he invented his RSA algorithm before anyone else did.

[cantab]

To me maths is fundamentally about proving things based on other things. You start with some "definitions" which serve to explain what you are talking about, and some "axioms" which are things you either think are obviously true or decide that for the sake of argument they're true. Then you work out what the consequences are.

The axioms are often really basic stuff. For example one of the "Peano axioms" of arithmetic states that x = x. Or one of Euclid's axioms of geometry is that you can draw a single unique circle with any centre and any radius. But from that kind of thing entire areas of mathematics can be built up and surprising and unexpected results proven.

In some branches of mathematics those definitions and axioms are based on real-world experience, and that usually makes the maths useful for solving real-world problems. But that's not required.

[FancyMouse]

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

Try this one.

[fredinno]

On 2/24/2016 at 2:04 AM, 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?

yo mama.