Mathematics; discovered or invented?

[Benzie]

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.

[owleye]

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

[Whut]

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.

[Watson]

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?

[Benzie]

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 :-) )

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...

[Jordan]

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.

[Lomax]

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.

[ComplexityofChaos]

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.

[owleye]

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

[dady551]

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

[Terry]

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.

[Don Juan]

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.

[RexAO]

I vote 'discovered' because if you had one stone and another stone, you will always have 2 stones, even if humans didn't exist.

[Pelargir]

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

[mitchellmckain]

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.

[Dave_Oblad]

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 :^)

[doogles]

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.

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’.

[mitchellmckain]

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.

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.

[Dave_Oblad]

Hi all,

Sorry Mitch, but while we may have the freedom to invent the symbols that represent various math functions, we can't make up or invent those functions.. as they represent absolute truths.

The difference between Euclidean and non-Euclidean geometry demonstrates this.

Are you kidding? That is absolutely worthless as a defense against invention. Both types of Geometry are based on very strict absolute rules that we most definitely had no choice in their invention.

You do understand that invention implies freedom of choice? Again, we may have such freedom in choosing the symbols that represent absolute truths but we can not invent the truths themselves.

Regards,
Dave :^)

[mitchellmckain]

Sorry Mitch, but while we may have the freedom to invent the symbols that represent various math functions, we can't make up or invent those functions.. as they represent absolute truths.

Sorry Dave, but you don't seem to be paying attention to what I said. The answers to specific questions in mathematics may be 100% determined, but the freedom in choosing which questions to ask is endless.

The difference between Euclidean and non-Euclidean geometry demonstrates this.

Are you kidding? That is absolutely worthless as a defense against invention. Both types of Geometry are based on very strict absolute rules that we most definitely had no choice in their invention.

You do understand that invention implies freedom of choice? Again, we may have such freedom in choosing the symbols that represent absolute truths but we can not invent the truths themselves.

Why would I make a defense against invention???

Of course, you don't seem to grasp the fact that our freedom of choice is in the set of rules we choose to follow and this is the difference between Euclidean and non-Euclidean geometries. Sure we all live in the same universe but that is just like playing many different games with the same pack of cards. The games can even be 100% deterministic but it doesn't change the freedom we have in choosing which rules to play by.

[phyti]

Math is a language. It expresses relations of abstract symbols according to well defined rules. The symbols represent things of interest and the operations performed. The expressions can be expanded as statements in a common language. The operations and elements are all by definition.

It began out of necessity to account for populations and properties, financial transactions, construction, and other human activities.

Knowledge is always incomplete and requires a process of constant refinement. What is discovered using math is what is true or false. Not all mathematical expressions are true, thus a quest for proofs.
Discovering truth is still a trial and error process.

[mitchellmckain]

The equal votes in this poll should suggest that this just isn't a black and white matter. Every discovery takes a little invention and every invention involves a great deal of discovery. Consider the invention of the light bulb, for example. did it not involve a considerable amount of discovery regarding the properties of all the materials which were tried for the filament of the bulb and the gaseous or vacuum content of the bulb? In all instances of discovery and invention there are both choices to make and unalterable truths to uncover.

[JustAsking]

I thought I'd revive this thread because it discusses something I've been pondering lately. But let me ask it a different way. Why does 1 + 1 = 2? Someone mentioned, well if there's 1 stone, and then there's another stone, you have 2 stones regardless of whether humans are around. But to me that's just putting a human definition on this symbol + . What about this: If you have one raindrop that falls to the ground. Then another falls right on top of it, how many raindrops are there? Obviously 1. So 1 + 1 = 1. So the meaning of the symbol + depends on the context. And to my mind that means math is just invented not discovered. Doesn't mean it's useless obviously, but let's recognize it for what it is: a highly evolved set of conventions that may or may not describe underlying reality.

Where this becomes interesting is in the notions of beauty in physics. And the talk is that since the math is beautiful, reality must be beautiful as well. But maybe the math is beautiful only because we're baking into the conventions we create. So this to me supports Sabine Hossenfelder's idea that beauty leads scientists astray.

Thoughts?

[davidm]

Stuff like this always brings to mind Borges’ Tlön, Uqbar, Orbis Tertius, in which, among many other strange things, we are encouraged to believe that nine equals one.

Suppose we lived in a world with radically different physical properties, such that, for example, when two objects came in close proximity, a third object materialized. Could it be said of such a world that one plus one equals three? If so, could a self-consistent arthiemtic and maths be derived from this?

This also brings to mind a chapter from the book “Beyond Experience: Metaphysical Theories and Philosophical Constraints,” by the philosopher Norman Swartz. The relevant passages are here, the topic being that we should not assume, as is so often done, that we ourselves, and intelligent aliens, could use mathematics as a basis of communication:

Must one have a concept of mass, for example, to do Newtonian mechanics? We might at first think so, since that is the way it was taught to most of us. We have been taught that there were, at its outset, three ‘fundamental’ concepts of Newtonian mechanics: mass, length, and time. (A fourth, electric charge, was added in the nineteenth century.) But it is far from clear that there is anything sacrosanct, privileged, necessary, or inevitable about this particular starting point. Some physicists in the nineteenth century ‘revised’ the conceptual basis of Newtonian mechanics and ‘defined’ mass itself in terms of length alone (the French system), and others in terms of length together with time (the astronomical system).8 The more important point is that it is by no means obvious that we would recognize an alien’s version of ‘Newtonian mechanics’. It is entirely conceivable that aliens should have hit upon a radically different man- ner of calculating the acceleration of falling bodies, of calculating the path of projectiles, of calculating the orbits of planets, etc., without using our concepts of mass, length, and time, indeed without using any, or very many, concepts we ourselves use.


Their mathematics, too, may be unrecognizable. In the 1920s, two versions of quantum mechanics appeared: Schro ̈dinger’s wave me- chanics and Heisenberg’s matrix mechanics. These theories were each possible only because mathematicians had in previous generations invented algebras for dealing with wave equations and with matrices. But it is entirely possible that advanced civilizations on different planets might not invent both algebras: one might invent only an algebra for wave equations, the other only a matrix algebra. Were they to try to communicate their respective physics, one to the other, they would meet with incomprehension: the receiving civilization would not understand the mathematics, or even for that matter understand that it was mathematics which was being transmitted. (Remember, the plan in S E T I is to send mathematical and physical information before the communicating parties attempt to establish conversation through natural language.) Among our own intellectual accomplishments, we happen to find an actual example of two different algebras. Their very existence, however, points up the possibility of radically different ways of doing mathematics, and suggests (although does not of course prove) that there may be other ways, even countless other ways, of doing mathematics, ways which we have not even begun to imagine, which are at least as different as are wave mechanics and matrix me-chanics.

[TheVat]

Seems like such questions can lead one into linguistics, i.e. asking just what it is that symbols are about, how they refer to a state of affairs - for example, asking if a world where bringing one object next to another results in three objects instead of two is really possible (speaking in basic arithmetic terms - I realize sexual reproduction leads to 1+1 = 3 all the time in our world) We are only able to ask the question in a way that makes this possibility seem incoherent and perhaps absurd. A third object, however it arose, would still be a third object. Numeracy seems basic to anything biological. If I eat one vlorg, and then I eat another vlorg I have eaten two vlorgs. If they make a third vlorg in my stomach, then I take an alien antacid, and I still retain a numeracy that allows me to differentiate between 2 and 3. Maybe I just space out my vlorg intake, so they stay at separate locations in my digestive tract.

Worlds that don't consist of distinct objects might not find arithmetic useful. But I doubt there would be any life to assign usefulness to any system of symbols. It's very unlikely that such a world would develop radio telescopes, so they aren't going to be among the worlds that get our strings of primes that we are trying to communicate our intelligence with. Sure, those alien SETI scientists could have some differences in their more exotic physics maths, but it's hard to coherently develop a scenario where they don't know what primes are, or a Fibonacci series, or hexagonal numbers, etc.

I agree with Bee (Sabine H.) that beauty can lead people astray. Our neural networks are wired to appreciate symmetry and other particular mathematical qualities that give pleasure, but that doesn't mean fundamental physics couldn't be a sloppy mess in some respects.

[Brent696]

JustAsking » October 17th, 2018, 10:34 am

I thought I'd revive this thread because it discusses something I've been pondering lately. But let me ask it a different way. Why does 1 + 1 = 2?

We tend to think of the Universe as a build up, something created in the same sense we might create something by taking units and adding them together.

But consider the Universe came into existence by division, first take 1 unit, and then begin dividing, again and again, then Math becomes the order of division, it mirrors the division by which the universe came into existence, and when all the math is done, so it returns to 1.

And 1 is the number of infinity.

[JustAsking]

But we don't have to get fancy talking about other worlds, etc. Just think of my simple example. Two raindrops hit the ground in the same spot and form a single larger raindrop. (Or in the air as they fall.) If + refers to counting, then 1+1=1 in this case. If it refers to measuring volume, then 1+1=2. So it depends on the context. We could form a mini mathematical law about raindrops using only + . So if we were developing a theory of raindrops, and we did experiments, we could "model" the results to get 1+1=1 or 1+1=2 depending on how we define +. I.e. we could fudge the math the get the results we want. And that's what I'm wondering if theoretical physicists are doing, only with math that's so complex they actually miss the contexts. If you read Sabine's book you see how hundreds of papers come out positing all kinds of ways that this complex math seems to lead them to just about any conclusion they want. And even saying there has to be a tie in to experimental results doesn't seem to be enough.

[Brent696]

JustAsking » October 19th, 2018, 10:04 am

But we don't have to get fancy talking about other worlds, etc. Just think of my simple example. Two raindrops hit the ground in the same spot and form a single larger raindrop. (Or in the air as they fall.) If + refers to counting, then 1+1=1 in this case. If it refers to measuring volume, then 1+1=2. So it depends on the context. We could form a mini mathematical law about raindrops using only + . So if we were developing a theory of raindrops, and we did experiments, we could "model" the results to get 1+1=1 or 1+1=2 depending on how we define +. I.e. we could fudge the math the get the results we want. And that's what I'm wondering if theoretical physicists are doing, only with math that's so complex they actually miss the contexts. If you read Sabine's book you see how hundreds of papers come out positing all kinds of ways that this complex math seems to lead them to just about any conclusion they want. And even saying there has to be a tie in to experimental results doesn't seem to be enough.

Sorry, you seem to be making an exercise in categorization, 1 rock plus another rock equals 2 rocks. 1 .05 ML raindrop plus another .05 ML raindrop might make a larger .01 ML raindrop, but in volume.

To determine or predict outcomes we apply math, set values to defined units, then add or subtract, etc... We are inventive in HOW we use math, but the underlying realities or principles are there beneath the surface of the Universe. These principles, laws, or patterns underlying reality have been discovered, then symbolized by scratches such as 1 or 2, and then these scratches are arranged into formulas that mirror those underlying patterns.

Math, as far as the schematics we use to communicate patterns, is invented, and seeks to express those Patterns that are discovered.

[hyksos]

Six years later, and it's still a 11/10 near exact split on votes.

[Brent696]

Six years later, and it's still a 11/10 near exact split on votes.

Perhaps one is yet unspoken, hiding behind a "like", and so do entangled pair make up the universe.