Is Mathematics There To Be Discovered Or Invented?

[JustNotCricket]

This is a question that I have pondered for some time.

Which numbers can be found in nature?

I raised a question two months ago asking -

Who invented binary numbers?

This led to the answers that I wanted and also to an interesting thread about the nature or lack of it of mathematics. So, I thought that I would put forward this question. If you are interested in the idea I would encourage you to look at the thread about Who invented binary numbers? which should not be too difficult to find. But please, do not continue that discussion on that thread. Start the new discussion here. Thanks.

So, this thread should be about whether mathematics exists and is to be discovered or whether it is an invention. If some mathematics exists and some must be invented then the thread may be about what exists mathematically and where can it be discovered.

And before we begin...
I hope that you enjoy this.

[jomifl]

Quantum mechanics presumably existed before i was invented. There could be a completely different way of understanding those relationships in nature so far only explained using maths. Maths just suits our brains because we invented it. We are so used to using maths to explain things that we have have almost forgotten that primitive man had his own way of understanding things without recourse to maths. Perhaps that is where gods came from :-)

[vascop]

@jomifl
QM didn't exist before complex analysis, but it certainly existed before you were invented.

[jomifl]

Trés drole vascop, So quantum mechanics suddenly appeared? maths has a lot to answer for then.

[JustNotCricket]

QUOTE "So, this thread should be about whether mathematics exists and is to be discovered or whether it is an invention. If some mathematics exists and some must be invented then the thread may be about what exists mathematically and where can it be discovered."

This is utter gibberish. Please rewrite in good English.

@methyl and ALL

Now, I can see how thoughts and discussions are going I will try and rephrase my question in two gibberish-free parts:

1 Which numbers can be seen in nature?
- this is clear enough I think
2 Are any mathematical operations visible in nature?
- by which I mean - Are there any examples of nature taking a set of numbers and operating on them?

[JustNotCricket]

@vasacop Thank you for this suggestion. I enjoy accessible maths books so I will give this one a try. I do not know whether it will begin to give answers to my questions but I am sure that it will offer or provoke a few more questions.

[jomifl]

You can see any numbers you like in nature, providing you have a brain that is appropriately programmed.
Pigeons can count up to seven but differential calculus is probably beyond them. They do a good job of flying though without the benefit of aerodynamic theory.

[jim360]

There's an old joke that does the rounds, of silly exam answers given by students presumably in a hurry to answer the question without thinking about it. One of my favourites is:

"Isaac Newton invented gravity."

But if maths is invented -- and certainly the maths Newton used to describe gravity is nowhere near the final picture -- perhaps this student was smarter than he or she gets credit for!

[JustNotCricket]

I see numbers in nature such as four legs on a horse, eight legs on a spider. Some might argue that each pair of legs is a totally different pair and others might counter with the fact that each pair is an attempt by the creature to create two identically similar legs in each pair. The number 2 is evident (to me at least). The number 1 can be given to pretty much anything. The number 0 might just be existing in the parts of space where there is nothing but I can see that this could start an argument which is not what I want. I think that 0 is better seen in the obvious idea that whilst in some places there are horses in other places we can truthfully state there are no horses.

Where are other integers observable in nature?

Can the three in three-dimensionality be considered as an observable integer? (Is the idea of having a third eye one in which we can see into a fourth dimension? - A simple yes/no answer to this question please, because it is not an area that I am particularly interested in).

Does anything in nature act in the way a mathematical function works by taking/recognising a number and giving a predictable output (not necessarily a number depending upon the number it takes/recognises? This is an idea that is suggested by the behaviour of ants mentioned earlier in this thread. Thankyou @WHOEVER SUGGESTED THIS.

And so, to the pub! Just hoping (before I go) that this continues to draw in thoughts, ideas and comments. I know it will reach an end at some point, and I do not know how it will reach a wider audience without restarting it with some improved question to kindle interest. Well, lets see.

[jim360]

In the cases you listed you aren't really seeing a number. You're seeing legs. That there happen to be four, or eight, or two, or six, or however many legs centipedes have, doesn't mean that you are seeing the numbers. Indeed, there are some tribes (I think, I vaguely remember this from some TV show) that don't even recognise numbers higher than three. So that their count would go "one, two, three, many." Another tribe has no words for numbers at all:

https://en.wikipedia.org/wiki/Pirahã;_people

Things are either "few", or "many". I think this does suggest that maths was invented to suit our needs, or the needs of our ancestors -- because if those needs (which are going to be related to farming) never exist, then neither it seems do the numbers or the concepts to go with them.

I'm not enough of an anthropologist to know more about this, about the development of languages and cultures. But it seems worth mentioning.

The more I think about it, the harder it is to say that maths was discovered. Sometimes the invented concepts may lead us to places we would never have considered. But even then that discovery is going on inside our heads.

[JustNotCricket]

@jomifl Whilst I do not know the experiments that have proven this fact, I assume that the pigeons you refer to must be counting. And what they are counting is seven ones. As for differential calculus - ask how a cricketer catches a ball and I would be surprised if calculus is mentioned. Counting and calculus are things that we have invented. But, if a pigeon can count, what does it do with the information? Does it start throwing extra eggs out of its nest? If this is the case then we might just have an example of a mathematical operation.

@jim360 Good old Newton. If only he had patented gravity then the science teachers of other nations would respect it and him a little more.

Perhaps Newton did invent gravity.
Consider this -
What if Newton got it wrong and invented something that was like gravity but was not exactly gravity itself? Maybe he missed out or added something. Could we then say that he had invented something? How would this be different to inventing something accurately?

[JustNotCricket]

@jim360
Quote
In the cases you listed you aren't really seeing a number. You're seeing legs. That there happen to be four, or eight, or two, or six, or however many legs centipedes have, doesn't mean that you are seeing the numbers.

Yes but this provokes two ideas in my head.
The first I have already suggested our two arms are two similar objects.
Secondly, the creatures themselves do not necessarily "see" the legs but they still have them and somehow they grow them in the same numbers each time a new creature is produced. Spiders are not seeing eight legs and nor are they thinking 8 (I assume this is true because I am fairly sure that baby humans do not spend anytime at all thinking "ah, I sense I have one leg and I really could do with two" they just have two legs but it is two legs that they have. And spiders, well, they have eight legs. (Therefore I believe that the number 8 is a number that "exists" in nature.)

[mibn2cweus]

Jake, I think ^ that's ^ your cue ;o)

[jomifl]

JNC When I said that pigeons can count up to seven I wasn't implyng that they have a concept of number and have a 7 based numbering system. What they can do is recognise the difference in quantity between the numbers of objects up to seven. How they do that is anybody's guess.

[mibn2cweus]

//Things are either "few", or "many". I think this does suggest that maths was invented to suit our needs, or the needs of our ancestors -- because if those needs (which are going to be related to farming) never exist, then neither it seems do the numbers or the concepts to go with them. //

Maths was invented by someone who found a spare moment in the fight for survival in order to create more spare moments . . . to do maths.

[jomifl]

Spiders have eight limbs that are only classified as limbs in our heads. In nature all the limbs are all different so only exist as single objects.

[Peter Pedant]

Blimey Jim !

Jim - who has just done a Cambridge course in Advanced Adding and Other Things didnt do a course in this

even tho it was Turing who sudduv started it off ( = Cambridge man )

Two camps for mad mathematiicians

The majority 80% are in the one that assents to [voice from ground] ' Hello I am a little lonely theorem waiting to be discovered.'

and the others - of which Brouwers was the main proponent - (Dutch) : that it is a system of symbols and you have rules and make combined statements (sentences have a technical definition) ( see Jim's post you have a set of objaaaaze (real numbers) and then another set of symbols (+,-,*, /) and then define what the effect of one on the other is. )

and then and then you can define a third set

[Peter Pedant]

oops - and then and then the interesting bit (!!!)
you can define upside down A (for all)
and upside down E (there exists)

and that allows you to make statements about your language using symbols which are not in that language ( = metastatements)

Jim - are you sure you didnt do a course in this ?

Post did somethng in this and so I think did Markov and Wamg....

[ludwig]

Before we invented maths there were just things. Lots of them. Some bigger than others, some far away and some near, some we could pick up and some we couldn't, some moving around and some staying still.
We invented maths to make sense of all that, because being able to make sense of it gave us an evolutionary advantage.

If some catastrophy wipes out human life, there won't be Maths any more. there will just be things again.

[JustNotCricket]

Thank you to everyone who contributed to this question. I will close it here and maybe I will open a similar discussion later on. There is no best answer but many ideas have been put forth and I for one have enjoyed thinking about what people have said. Thank you all again.