Mathematics ?

[topdude]

This might be a strange question and I'm not even sure how to ask it !!

What is mathematics ?

Is it an integral part of the universe that us humans have "discovered " ?

Or is it something we have "invented " to explain what we observe ?

Does that even make sense ?

[Alien 13]

Good question, my dog knows nothing about maths but can catch a ball in flight so I expect the answer is the first one.

Alan

[Stu]

Good question. I would say the latter, because the universe would operate whether maths existed or not.

Alan's dog has just observed and learned, cleverly bypassing the need to learn maths [emoji3][emoji106]

[Stub Mandrel]

Maths is an abstraction that allows us to describe the innate properties of the universe.

It's very hard to be sure if concepts such as unity, zero, integer or equality have any absolute meaning? At the level of quantum mechanics the idea of a change of exactly 1 is troublesome as it has a probability function attached to it.

Ow, scary. I go with the Alan's Dog Theory.

[pipnina]

Good question. I would say the latter, because the universe would operate whether maths existed or not.

Alan's dog has just observed and learned, cleverly bypassing the need to learn maths [emoji3][emoji106]

Or, Alan's dog has become the world's greatest mathematician / physicist, and calculates the velocity of the ball and where it will land in order to intercept it perfectly.

In my college course, my lecturer for the matsh unit gave us a presentation on "What is maths?". I can't remember his main point from it but it gost a little conspiracy theoristy at points (he was pointing out things like how the fibonnaci numbers are everywhere & how doing something to a river always results in PI...

[George Jones]

This subjective, so there is no one accepted answer (even by professional mathematicians). My answer (thus my opinion) is "discovered". There are mathematicians in both camps. British mathematician G.H. Hardy also was in the "discovered" camp, and wrote

"... and there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is 'mental' and that in some sense we construct it, others that it is outside and independent of us ... I believe that mathematical reality lies outside of us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply our notes of our observations."

[Ouroboros]

Is maths the way we do the accountancy of quantities that the universe does naturally?

[Paul M]

My pet idea is that it's a kind of language and therefore a human construct and I don't just mean in its written form but conceptually too.

Mathematics might perfectly describe the properties of spacetime and allow us to predict what happens to an orbiting body.

But the orbiting body doesn't need maths, it just follows the shape of the space in which it exists.

It's the same with all the physical properties of the universe. They know what to do because the universe won't fit together any other way. We just invented a tool to model it.

I once read some stuff about "tiling the plane". Fascinating, but I couldn't help think that it wasn't really a maths problem but maths was useful in probing the subject.

Anyway, I knew maths was for losers when I gave up on learning the 2x table

What possible use would that have?   

[Size9Hex]

My tuppence worth:

Mathematical entities are an integral part of the universe that we discover. The notation we use in relation to those entities is invented.

The Mandlebrot Set to me is a fantastic example of a discovery.

There are so many mathematical entities that crop up time and time again (prime numbers, e, pi, phi to name a few) when you're not even looking for them, that I couldn't see how they were invented. They're just "right". They exist in isolation of our own ways of denoting the mathematical universe. Aliens on the other side of the universe will discover these entities too, although their notation for them will be different.

Our notation/language for of writing/conveying the entities is not a discovery, but an invention that can help or hinder us. Base 10 numerals are a good example. A very powerful notation, but to my knowledge based purely on how many fingers we have. No reason why another species would use 10 instead of some other number, or some other system entirely. The Roman system is quite different and is a comparatively rubbish invention in terms of facilitating any understanding.

[symesie04]

Interesting.

At first i thought well maths surely has to be a creation of humans to help explain and predict the environment in which they find themselves. But then i thought but wait if so then what are the chances that a notion we have created would scale up and fit the enormity of the Universe "almost" perfectly? Surely if the physics that define our Universe didnt follow a predefined mathematical pattern then it would be purely random and if so than no mathematical pattern we create would scale across it? Therefore physics must follow a pattern of sorts and we simply apply our definition of it in the form of mathematics.

The only problem with that is that if physics follows a predefined pattern then does that suggest a grand design? eeek.

[Macavity]

Me? I'd revisit the 1st Chap. of the "Road to Reality" (Roger Penrose)

He discusses the THREE worlds: Platonic *mathematical*, physical and

mental (and their relationships) far better than I ever could even begin...

Aside: If anyone is wary of trying to pursue a career in Science because

they are "no good a maths"? Do you best! Get good results in science.

I'm not saying that the "Philosophy of Science" is not interesting - Just

that so many today seems to be obsessed with... "understanding why"

and the "purity" of things? Science needs people who can *do* stuff...

[JB80]

The Dave Mermin approach.

[Stub Mandrel]

Aside: If anyone is wary of trying to pursue a career in Science because

they are "no good a maths"? Do you best! Get good results in science.

My daughter hated maths and said she couldn't do any of it; all I could do was make here more confused. She was home educated and my wife arranged an hour a week with another home eductaor who is also a teacher - result was an A at GCSE and she picks me up when my mental maths is out :-/

[Tiki]

What is mathematics ?

A good question with no clear answer.

Personally, I believe that mathematics is discovered rather than invented.

If mathematics was purely a human invention then we might expect some differences within the subject on purely cultural grounds. The very fact that Pythagoras's theorem (amongst others) has been independently discovered many times by different people with hugely varying socio-economic backgrounds is a strong argument against inventionism.

[Paul M]

I wonder if some of the difference between the two camps is rooted in semantics?

Take the example of Pythagoras Theorem.

In my mind Pythagoras Theorem is just a mathematical tool, an invention using our universal language of mathematics, to explore the purely geometrical relationship between the sides of a right triangle. It only holds true across cultural, intellectual and language barriers because we created a language to transcend those boundaries. 

I don't need any algebra or other notation to visualise what's happening with Pythagoras or Pi for that matter. These things just "are". They are defined by the geometrical properties of space. They don't need defining by maths to "work".

So everything that mathematics represents is natural, properties of the universe just waiting to be discovered. These thing were discovered. Mathematics is an invention we created to discover and explore them.

I said somewhere in an other thread that everything in our universe must be definable and predictable by maths but not everything that maths predicts is compelled to exist in nature.

I think...

[ollypenrice]

Galileo: Mathematics is the language in which God has written the Universe.

Me: Mathematics is the language in which we read the universe.

Personally I think that there are two universes, the one which is (and which may be partially or entirely incessible to us) and the one we read. If we read it in the language of mathematics then perhaps that's because mathematics is the language in which we wrote it. We could hardly be expected to write it in a language we didn't speak...

Olly

[Martin Meredith]

Isn't there a problem here with talking about mathematics as if it were a single thing? There are lots of branches of maths, some of them which are certainly invented without any need to describe any kind of external 'reality'. I would say a lot of the mathematics of cryptography falls into this class. I doubt that there is a part of the universe that needs describing in terms, say, of the mathematics underpinning the Enigma Machine. At its most prosaic, there is nothing to stop any one of us inventing our own branch of mathematics, complete with axioms and laws, without any need for it to represent anything at all except itself.

On the other hand, there do appear to be physical laws that can be described in mathematical terms. In these cases I'd say it makes sense to think of discovering the law and representing it with mathematics. Not that there is a 1-1 relationship between laws and mathematics, since the same bit of maths can be used to describe more than one law.

As for the dog, who knows what kinds of representation they have to help them catch a ball? A related example is the way bees take off and land. There is strong evidence that bees judge speed based on the rate at which the texture of the land is rushing past them, relatively, and that they aim to keep that speed constant, so that when they are in the air they can fly faster, but when coming in to land they have to slow down to maintain a constant 'texture speed' given that the ground is much closer. There are some scientists (in Brisbane I think) who have used the same principle to fly aircraft autonomously.

So the bee seems to have learned a simple law of perspective about the way objects look smaller the further they are from them, and turned it into perhaps the most simple operational law. 

Martin

[ollypenrice]

Once we start imposing our mathematics on the dog and the bee we start describing one of our understandings of  what they do. But we have several ways of understanding what they do. (Martin's various maths, above.)  We can perceive geometry visually and so, I dare say, can the dog and the bee. If we see two propagating and converging lines we can 'see' where they will intersect. Surely it is just plain wrong to say that we have performed this or that order of calculation in order to arrive at this. I think we, the dog and the bee can predict the intersection without either algebra or arithmetic, though algebra and arithmetic can also predict the intersection.

I don't understand the relationship between mathematics and reality but it seems that there is one. (At least between our reality and our mathematics, which might ring an alarm bell...) But mathematics is systematic, by its very nature. And if the universe (or our perception of it) were not systematic then surely it would be no more than noise without signal? In which case we would not, in any meanigful sense, exist at all. We would certainly be in no position to describe anything, let alone reality. So any reality which is describable to those within it must be systematic, and maths is systematic, so at least there might be a hint, there, as to why mathematics works for us?

Olly

[Stub Mandrel]

Is maths intrisic to the universe. I think it probably is.

When poop scooping today, the process of inverting the bag made me think 'how do we define when something is inside?'

Perhaps when it passes through the entrance?

But to do that with certainty, thinking topographically, you have to loop round and pass through twice.

So being inside and outside a bag is meaningless... but you can be inside or outside a sphere.

What about tying up the bag? How do we define whether or knot(LOL!) the ends of the bag are tied together rather than just tangled up?

Enough!

I hot-footed it to the poop bin and abandoned the experiment.

[Ruud]

Mathematics is a formal system of axioms and theorems. Theorems must be proved, axioms are assumed to be true. Mathematics has many branches. Most of us know a bit of calculus and algebra.

Mathematics is a discipline which studies itself. It is man-made. It's not a science, but science can use it.

In Dutch it's called wiskunde, the knowledge of knowing/being certain.

A nice book on mathematics is The Outer Limits of Reason by Yanofsky.

[pipnina]

Is maths intrisic to the universe. I think it probably is.

When poop scooping today, the process of inverting the bag made me think 'how do we define when something is inside?'

Perhaps when it passes through the entrance?

But to do that with certainty, thinking topographically, you have to loop round and pass through twice.

So being inside and outside a bag is meaningless... but you can be inside or outside a sphere.

What about tying up the bag? How do we define whether or knot(LOL!) the ends of the bag are tied together rather than just tangled up?

Enough!

I hot-footed it to the poop bin and abandoned the experiment.

Have a look at klein bottles... An object which I don't think actually has an inside or an outside... https://upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Klein_bottle.svg/2000px-Klein_bottle.svg.png

[Stub Mandrel]

Have a look at klein bottles... An object which I don't think actually has an inside or an outside... https://upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Klein_bottle.svg/2000px-Klein_bottle.svg.png

I'd forgotten those, but I suspect they were in the back of my mind.

A great subject for 3D printing, or glass blowing.

[topdude]

Wow ! Some really interesting replies to my original question.

I suppose what made me think about this is I understand that you can make an observation then " do the maths" in order to explain what was observed.

But how the hell do you " do the maths " first then make the observations to prove or disprove the calculations ?

This sort of makes me feel that mathematics is behind everything and was there to be discovered !!!

[vlaiv]

This is really interesting question.

Let me give you my take on it and possibly expand discussion.

Depends on semantics of the word mathematics. If one uses the word to describe set of symbols we use to communicate concept (of mathematics), i.e. the language then it is invented. If we look at the concept of mathematics I believe it is discovered.

Further more, chicken&egg sort of question pops into mind - which came first, the universe or mathematics? Is mathematics inherent property of the universe, or is universe byproduct of mathematics?

I believe it is second one - universe as byproduct of mathematics. Let me make following argument for that one:

1. Existence as opposed to nothingness (evident)

2. The furthest abstraction of mathematics is as follows: existence of entities, existence of relationships

3. Same goes for universe (as in point 2)

4. Isomorphism between the two abstractions (entity, relationship)

5. Mathematics has the property of being self-consistent.

From above one might conclude that mathematics and universe have same underlying structure. I tend to give mathematics the slight edge due to property of self-consistency that is more evident in mathematics than in universe

Does any of this make any sense?

Vladimir

 

[Kishen20]

I didn't get about half of this thread (blame me, not this thread) and therefore skipped (so I may missed something important), but isn't Mathematics used to explain the Universe, technically?
We humans need metres for distance, degree Celsius/Kelvin for temperature, seconds, minutes and hours for time (and angles) etc etc, so why not Maths is used to explain the Universe?
Some other intelligent being out there in space may use another method to explain what's going on around it, and it may be correct too, at least in their way.