Mathematics; discovered or invented?

Philosophical, mathematical and computational logic, linguistics, formal argument, game theory, fallacies, paradoxes, puzzles and other related issues.

Is mathematics discovered or invented?

Discovered
10
56%
Invented
8
44%
 
Total votes : 18

Mathematics; discovered or invented?

Postby Benzie on February 27th, 2012, 10:17 am 

Is mathematics discovered or invented?

It seems a straight forward matter of opinion, but if we stop and think more about what it is really asking there comes to light an interesting debate.

If it is discovered then we can argue that it is there regardless of human activity. This suggests mathematics is some objective and abstract structure, which exists regardless of humanity and can only be uncovered by 'searching' for it.
However, the notion of it being invented also holds some weight, after all, we can quite easily understand what someone means when they say Cantor invented Transfinite mathematics. He worked on it, he constructed the equations and so on.

Obviously these views are just the tip of the iceberg, I get the feeling there is a lot more to them than I can outline in a brief paragraph.

D.
Benzie
Forum Neophyte
 
Posts: 15
Joined: 01 Sep 2011


Re: Mathematics; discovered or invented?

Postby owleye on February 27th, 2012, 12:27 pm 

It appears what you have in mind for the debate is to have it centered on the idea that these two concepts are mutually exclusive and such that mathematical objects have to be one or the other. Then, because, for example, we or a computer can be said to discover them (say some number having a certain property) that hadn't been known before, mathematical objects must be discovered, not invented. But you caution that this (discovery) implies that these objects must exist absent human existence. And this should be grounds for some pause. On its face, though, one shouldn't draw that conclusion. All it means is that such a discovery implies following certain rules and procedures. As hinted, one can use a computer to find them. So the question devolves not to the mathematical objects themselves, but rather to the rules and procedures that are set up to finding them. Are these invented or discovered?

James
owleye
Honored Member
 
Posts: 5699
Joined: 19 Sep 2009


Re: Mathematics; discovered or invented?

Postby Whut on February 27th, 2012, 1:30 pm 

Invented. I'd say it's analogous to language. We create words to describe specific experiences, can use these words to form sentances, which aid us in further understanding. In the same way we create numbers to describe specific experiences, can use these numbers to form sums, which aid us in further understanding. One might argue 1+1=2 regardless of our existence, so there is: the inner workings of math we dicovered, but there's no 1+1 without our existence. I suppose we do discover the answers to our sums etc, but invented the method of finding them.

Perhaps, it's the purest of inventions. The best way of representing reality we've ever created.
Last edited by Whut on February 27th, 2012, 2:03 pm, edited 3 times in total.
Whut
Active Member
 
Posts: 1065
Joined: 10 Sep 2010


Re: Mathematics; discovered or invented?

Postby Watson on February 27th, 2012, 1:43 pm 

I would think this is the same question as asking, " Did Columbus discover or invent the new world?" It was there to be found. Math works and is there to be found? But like all knowledge, it is a learning what works process, to build on with new ideas on top of past ideas.
So, I'm going with discovered?
User avatar
Watson
Resident Member
 
Posts: 4382
Joined: 19 Apr 2009
Location: Earth, middle of the top half, but only briefly each 24 hours.


Re: Mathematics; discovered or invented?

Postby Benzie on February 27th, 2012, 2:43 pm 

owleye wrote:So the question devolves not to the mathematical objects themselves, but rather to the rules and procedures that are set up to finding them. Are these invented or discovered?


Thanks, you pointed out a mistake there - assuming I'm reading it correctly.
I was referring to mathematics as a set of principles, not the objects (which I take to be numbers, shapes etc - is that right?). So perhaps I should have asked; 'are the principles of mathematics, the laws and so on, out there waiting to be discovered? - or do we have to invent them?' - and yes, assuming they are mutually exclusive :-p

On top of this, I guess it's possible that you could argue this is more accurately a question about Logic (very Fregean i know, but Im a fan) and whether or not the principles of logic are really out there waiting to be discovered or do they need inventing.
I suppose this depends on what is meant by out there... some abstract, intellectual wilderness perhaps? (make sure you get the right immunisations before going to explore :-) )

Whut wrote:the inner workings of math we dicovered, but there's no 1+1 without our existence.


agreed, i think that is again the distinction between mathematical objects and the principles...
Benzie
Forum Neophyte
 
Posts: 15
Joined: 01 Sep 2011


Re: Mathematics; discovered or invented?

Postby Jordan on February 27th, 2012, 6:40 pm 

Extremely limited as my knowledge is I would say if pressed that mathematics is an invention rather than a discovery, I perceive maths as a language and a tool. Knoweledge of mathmatics seems present among the animal kingdom too?

The thread reminded me of a documentary series by the BBC first aired a few years ago called The Story of Maths which was in four parts, each an hour long. I've included the link to the first part and you'll find the other three in the links.

User avatar
Jordan
Member
 
Posts: 377
Joined: 19 Nov 2010


Re: Mathematics; discovered or invented?

Postby Lomax on October 3rd, 2012, 11:04 pm 

I voted "discovered", for the same reason given by Watson, Stanley Huang and others. No matter how hard we try, we can't make it false that 1 + 1 = 2.

The proponents of mathematical invention tend to argue that we created all those symbols, which is true for sure, but try extending their argument to natural languages. Did we invent the universe because we invented "the universe"? Inventing the symbols and inventing the thing, are not the same. If numbers are nothing more than their symbols, or if we could have made 1 + 1 =/= 2, it remains to be shown.
User avatar
Lomax
Forum Administrator
 
Posts: 3428
Joined: 01 Jul 2010
Location: Nuneaton, UK


Re: Mathematics; discovered or invented?

Postby ComplexityofChaos on November 3rd, 2012, 11:24 pm 

It's an interesting question, which I have wrestled with a lot. The Incompleteness theorems would favor the discovered view, which is the opinion I used to hold. Now, I'm not so sure. I'm leaning in favor of both. There are often times more than one way to prove a mathematical "statement", and that being the case, I think mathematical proofs are at least partly the result of human invention. Same with theorems.

On the other hand, math certainly seems to also involve discoveries. The rate of increase of a sphere, as an example, that has to be a discovery, since that is how spheres behave in the real world. In such a case, it is like discovering a law of physics. So, we know there is a great deal of math that involves discoveries, and makes math a real science, like in statistics, probability theory, geometry, etc.

But then, what about areas of math where there is no physical realm being described? In such a case, it is hard to view math as a discovery, and at those times it seems to be an invention.
User avatar
ComplexityofChaos
Banned User
 
Posts: 290
Joined: 03 Nov 2012


Re: Mathematics; discovered or invented?

Postby owleye on November 4th, 2012, 6:05 pm 

The topic began rather unfortunately, since it failed to tell us what we are to understand mathematics to be prior to indicating whether such a thing could be invented or discovered. This lapse was subsequently clarified somewhat by regarding it as a subset of logic, one perhaps that deals with numbers or spaces. In any case, it is supposed to reside within realm of analyticity in which its claims, derived though they may be, are true because they follow logically from proven claims, an axiomatic foundation that includes definitions of basic terms on which they are based. This, of course, leaves it a bit open-ended in that some claims are not known to be true or false because either they are demonstrated to be undecidable or not yet decided one way or the other. And, in addition, there are language elements, such as the requirement for distinct symbols, arranged in a certain way, spatially and temporally in order to demonstrate the proofs. Possibly such arrangements are are arbitrary in some sense, but it's difficult for me to think about such demonstrations in the absence of such spatial and temporal arrangements. Notwithstanding, some of it can be mitigated by way of the distinction between types and tokens, thereby, making it clear how software can be distinguished from hardware. However, if the content of all this can be distinguished from the language in which it is implemented, so to speak, reaching a level of abstraction that removes space and time altogether, one question arises, then, which is whether or not this abstract domain exists absent humans having these abstractions in their head, cloaked as they might be in spatial-temporal frameworks.

This is not the question of the OP, of course, but it outlines the sort of thing that is the subject of discovery or invention. Obviously, humans discover whether their claims are true or not (when they in fact do so). This shouldn't even be a matter of dispute (unless, of course, you have a view of knowledge acquisition that is as Plato subscribes in which we merely recall the truths from a prior life). The question then centers on whether or not such demonstrations are totally the result of an invented axiomatic foundation. Well, part of this as well is obvious. Clearly, the mathematics that were in common use existed (long) before the axioms and postulates and definitions on which they depend. As such, these axioms have to be considered as discovered.

However, it was only with the advent of a new propositional logic of Frege, that the field opened up in such a way that the fields of mathematics and logic came to be distinct in the way it is today from empirical science. (A century of mathematicians prior to this development, of course, led the way to such opening up.) The notion of discovery in mathematics has to be distinguished from discovery in empirical science. And this is so because mathematicians today assume a particular logical framework or foundation, either tacitly or explicitly in order to develop the challenges they wish to create for themselves. All their work, then, is confined to a realm (a universe of discourse?) in which its truths are determined. And such realms are arbitrarily chosen. It is in that sense that mathematicians can be said to work within a realm of their invention. It's rather an arena of supposition.

James
owleye
Honored Member
 
Posts: 5699
Joined: 19 Sep 2009


Re: Mathematics; discovered or invented?

Postby dady551 on July 10th, 2013, 10:40 pm 

if we look into the basic definitions of invention and discovery you will come to know that math is discovered not invented because it has been there since the beginning of time
User avatar
dady551
Member
 
Posts: 82
Joined: 08 Jul 2013


Re: Mathematics; discovered or invented?

Postby Terry on July 13th, 2013, 10:47 am 

Can we think of the universe as a physical presentation of an underlying real immaterial existence? An abstraction out of which instance of various intangible structures appear as different physical forms.That's why different phenomena might be described by the same equations. That's why we can find physical instances of pure mathematical objects. That's why we can abstract from diverse physical occurrences and derive the general pattern as mathematics. The symbolic representation of mathematics itself is a way of physical presentation of this underlying real immaterial existence. IT can be presented in various symbolic forms as different instances as well but is ITSELF presentation independent. We don't create the abstraction ourselves. They are there to be instanciated. So, in a deep sense, we don't create anything. We just follow the way the universe works.
Terry
Member
 
Posts: 439
Joined: 01 Sep 2008


Re: Mathematics; discovered or invented?

Postby Don Juan on July 15th, 2013, 11:21 am 

I think mathematics is partly discovered and partly invented. The relationship of patterns and the distinctions of such relationships have to include both discovery and invention. Discovery for noticing the patterns and invention for marking the patterns and completing and expressing the distinctions. Why is there no choice provided for both? At least it can have 1 vote from me.
Don Juan
Active Member
 
Posts: 1113
Joined: 17 Jun 2010


Re: Mathematics; discovered or invented?

Postby RexAO on July 18th, 2017, 12:46 pm 

I vote 'discovered' because if you had one stone and another stone, you will always have 2 stones, even if humans didn't exist.
RexAO
Forum Neophyte
 
Posts: 2
Joined: 18 Jul 2017


Re: Mathematics; discovered or invented?

Postby Pelargir on September 11th, 2017, 4:54 pm 

I was always of the opinion that maths were discovered and learning a programming language made it even more obvious for me. Even though numbers like 2 or 4 are invented, they are only names for constants. It does not matter whether you say the number of the cows on the field is two, deux, zwei, duo or anything else, there will always be 2 cows on the field
Pelargir
Forum Neophyte
 
Posts: 11
Joined: 11 Sep 2017


Re: Mathematics; discovered or invented?

Postby mitchellmckain on September 13th, 2017, 5:53 pm 

I know it seems like math is out there to be discovered, and at one time I would have answered that this was the case. Perhaps this is because more than any other science its truths are a matter of proof rather than simply suggested by the observed evidence.

But recently I have had considerable doubts about whether there really is anything inevitable about our mathematical truths. Sure they are a certain product of logical deduction, but only after we adopt some set of premises with which to start. We have already demonstrated in non-Euclidean geometry that starting with a different set of assumptions we can come to very different conclusions. In many ways it is similar inventing a game like chess or go and discovering what makes for the best way to play.

Another objection I have is whether any argument for mathematical discovery rather than invention can actually draw a fixed line between mathematics any any of the other things we would say are invented. Are not all our inventions largely dictated by necessity? Sure there are elements of creativity and arbitrary choice, but there is still a hard core of necessity beneath them as well. I see no reason why this same observation should not apply to mathematics also. The correct answer may be fixed but the derivation of that answer can take many paths and therein lies the element of creativity and choice.
User avatar
mitchellmckain
Member
 
Posts: 569
Joined: 27 Oct 2016


Re: Mathematics; discovered or invented?

Postby Dave_Oblad on September 14th, 2017, 2:53 am 

Hi All,

That which is invented has freedom.. such as language, art, music, laws, morality.. etc.

Math has no such freedom.. it is composed of Rules, Rules that have been discovered over time.

We may discover many ways to approach a single math problem, but we can't make up our own rules.

Regards,
Dave :^)
User avatar
Dave_Oblad
Resident Member
 
Posts: 3208
Joined: 08 Sep 2010
Location: Tucson, Arizona
Blog: View Blog (2)
RJG liked this post


Re: Mathematics; discovered or invented?

Postby doogles on September 14th, 2017, 4:50 am 

FWIW, I have a slightly different take on mathematics.

To my mind, mathematics is a human concept of abstract representations of one aspect of natural phenomena such as the two cows in a paddock. It seems to me more appropriate to say that the discipline of mathematics has evolved, rather than been ‘invented’ or ‘discovered’.

Obviously it's a discipline used initially for communication of the notion of numbers either in verbal or written form.

Stone age people such as the aborigines had no written language but they verbally communicated numerical data by pointing to fingers and toes (and other body parts for larger numbers). See https://en.wikipedia.org/wiki/Australia ... numeration .

An improvement on the use of body parts for counting evolved in the form of the abacus. See https://en.wikipedia.org/wiki/Abacus . It was an evolved improvement to indicate numbers. It could also be used for a variety of calculations. According to the link above, it has evolved over millennia - “The exact origin of the abacus is still unknown. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of wood, stone, or metal.”

In written form, the notion of multiples in Roman and Greek numerology was indicated by '1' up to the notion of the sum of '111', and other symbols began to be used (evolved) for larger numbers. I imagine that the work of expressing the notion of '100' as 100 separate repetitive digits resulted in the development of such a set of symbols in small bursts of advancements. For example, the Romans used ‘C’ for 100 and ‘M’ for 1000.

In one of these apparent small advancements, Arabic symbols appeared. There is a suggestion that may never be proved or disproved that the original 9 figures each had an equivalent number of angles corresponding to the number of anything it was meant to represent in every-day life. With centuries of use, these symbols may have become modified in an evolutionary manner into the more roundish symbols we scribble today.

NUMBERS ARE ANGLES.jpg


When the Indian contribution of the concept of a zero came into use in about the 9th century AD, all of the necessary ingredients were available for the evolution of decimals, algorythms, calculus, logarithms, calculators, computers etc.

So to my mind, mathematics has evolved in small advances over millennia. Maybe each minor advance could be called an invention. But rather than ‘invention’, maybe we could say ‘ínsight’, ‘brainwave’, ‘good idea’.
doogles
Member
 
Posts: 836
Joined: 11 Apr 2009


Re: Mathematics; discovered or invented?

Postby mitchellmckain on September 14th, 2017, 2:39 pm 

It has long been an assumption in science fiction films that mathematics is a universal language and ideal for the purpose of communicating with aliens. The more recent film "Arrival" questioned this assumption as I think is correct.

In the film, many different countries are trying to communicate with the aliens and the Chinese had some success using the game of Majong to do so. Like I suggested above, I am not sure that mathematics is really all that different from a game. All games are defined by a set of rules. And with time working with such rules we may well come to all kinds of inescapable truths within such rules.

But I think we make too many assumptions about aliens in the idea that mathematics is some kind of universal language. I can easily imagine them trying to communicate with us using what seems to us like some sort of crazy game with rules we have never even imagined.

Dave_Oblad » September 14th, 2017, 1:53 am wrote:Hi All,

That which is invented has freedom.. such as language, art, music, laws, morality.. etc.

Math has no such freedom.. it is composed of Rules, Rules that have been discovered over time.

We may discover many ways to approach a single math problem, but we can't make up our own rules.

Regards,
Dave :^)

But that is just the point. We most certainly CAN make up our own rules. The difference between Euclidean and non-Euclidean geometry demonstrates this. And then there are various branches of mathematics like modular arithmetic. Furthermore if you read the story of Ramanujan, a great Indian mathematician, there is strong indications that he also came up with a lot of his discoveries using his own set of rules, which would sound quite bizarre to the mathematics community.
User avatar
mitchellmckain
Member
 
Posts: 569
Joined: 27 Oct 2016



Return to Logic

Who is online

Users browsing this forum: No registered users and 3 guests