The above question has been a long-standing source of debate. ‘Discovered’ means that the structures of mathematics exist objectively and independently while ‘invented’ implies that mathematics is the free creation of the human mind. The same debate exists in science.
An article profiles a mathematician Sergiu Klainerman who is convinced that it is discovered.
The equations that govern black holes were true before there were black holes. That claim is hotly contested, and cuts through one of the deepest fault lines in the philosophy of mathematics. On one side are those who hold that mathematical structures, including well-established principles and basic geometric shapes like the tetrahedron, exist independently of human thought – not as a language we invented to describe reality, but rather as the substrate of reality itself. On the other side of the debate are those who argue that mathematics is the product of human labours, imposed on a world that would be wholly indifferent to it were we not here.
He started down this road after reading an essay by Eugene Wigner.
Early in graduate school, Klainerman came across an essay by the Nobel Prize-winning physicist Eugene Wigner – ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’ (1960) – that would stay with him for the rest of his career. Wigner’s subject was the uncanny way that mathematical concepts developed for one purpose keep turning up, unbidden, in other settings that seem to be wholly unrelated. As an example, Wigner told the story of two former high-school classmates who were discussing their professions years later. One, a statistician, showed his friend an expression for a term related to population density that depended on pi, the ratio of a circle’s circumference to its diameter.
‘Surely the population has nothing to do with the circumference of the circle,’ the friend replied. And yet it did. The mathematical formulations devised to elucidate specific aspects of the physical world reappeared with surprising frequency as accurate descriptions of phenomena they were never intended to address.
…What he wanted to write about, above all, was the mystery that had haunted him since he first read Wigner as a graduate student: why does mathematics work so unreasonably well and keep rearing its head, time and again, in the unlikeliest of places? Klainerman agrees with Wigner that this qualifies as a genuine mystery – deep, unresolved and woven into fabric that connects abstract thought, mathematical principles and physical reality.
…Klainerman had long been inspired by Wigner’s essay. But as his thinking evolved over the years, he came to disagree with the eminent physicist on one essential point – the definition of mathematics itself. Wigner described mathematics as ‘the science of skillful operations, with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts.’ Klainerman takes issue with the word ‘invented’.
Tools and methods can be invented, but the principal ideas are discovered. No one can claim to have invented pi – it is a fact of nature that the circumference of any circle divided by its diameter yields the same irrational number, starting with 3.14159 and continuing forever. Pi was discovered, not invented, though someone did have to invent long division to compute it. Similarly, Klainerman noted, ‘five plus five was 10, long before we were born.’
I find this point of view interesting and suggestive but not conclusive, partly because I think a strong case can be made that scientific theories are invented and not discovered and argue so in my book The Great Paradox of Science. But what is true for physical theories may not be true for mathematics and I do not know mathematics history well enough to arrive at any definite conclusion.
Everyone please take one side or the other, or we’ll have to deal with some horrible neologism like “discovented”!
Disco-vented? Isn’t that a mirror-ball mounted on a fan?
The side I will take is both. I am not a mathematician (obviously), it seems to me that there is room for both views.
The symbol or equation we use for a mathematical concept is obviously invented, but the underlying concept the symbol or equation describes may be a description of a reality which exists independently of the symbol or equation. Further, if we ever encounter aliens and their mathematical theories we may well find they describe the same underlying reality, but at the same time the symbols/equations they use might elucidate some facet which are poorly, or even not, expressed in the equations we have invented to describe the same discovery.
We do this with natural language all the time, although natural language is significantly more vague than mathematical equations. European humans invented the concept of a cereal before they knew the existence of maize, when maize was discovered the invented concept of cereal was found to apply.
Mathematics is a more precise language, invented to describe discovered things. But there is probably a delta-epsilon limit to even the mathematical language and not a precise 1:1 correspondence between the invented language to describe realty and the reality which existed prior to the discovery of that reality.
But I do see both sides of this discussion and see that each side has some interesting arguments in its favor.
It would seem to me that there is a way to test this. Try creating different systems of manipulating values and if you find at least one which is just as good or better at describing the physical universe than what we call math, and you’ll definitely know it was created. If no one can do it, then it’s probably discovered. I say “probably” to the latter because it might reflect limitations of the human mind rather than fundamental reality, though in that case we’d be forced to treat it as if it were discovered regardless.
Or if it is discovered and we got most of the basics right, wouldn’t it be funny if one or two mathematical principals/conventions were actually legacy errors created by past cultures which didn’t quite know what they were doing? And now due to systems inertia are too sacred of cows to even consider taking a closer look at, and we’re forced to use an overly complicated hack job forever.
It’s invented. Mathematics is a formal system, and its objects do not instantiate objectively. One chooses a set of axioms, which can be done freely up to coherence (that is, the axioms do not entail a contradiction), at which point the logical consequences of that set of axioms is locked-in and cannot be altered internally to the theory. However, there yet remains room here for discovery to occur, for the fact that the logical consequences of the axioms all follow of logical necessity, it is not always known or obvious what those consequences are, or how they can be derived. Consequently, we discover the theorems of the theory by means of proof.
I would say some of both. Pi is a universal constant and is going to turn up in any physical or mathematical system. 360 degrees of a circle is a human invention, an arbitrary base number for dividing circles. Various numbers may be more or less convenient but don’t have any deep significance.
You can also manipulate how you write math to change the appearance of things but this doesn’t change the underlying reality. Doing your math in radians means seeing Pi less often. You can write any formula without Pi by using other constants. This doesn’t change the basic circle ratio.
Speaking of Pi, that’s actually one of the areas which could be a sign of an over-complicated hack job: Pi is the ratio of the diameter to the circumference (under the special case where the spatial curvature is zero, but let’s ignore that), yet standard geometry and trigonometry use the radius for everything instead of the diameter. At the very least, that makes some things less intuitive for no good reason.
(In case anyone thinks to bring up the point that there are multiple infinite series which converge on Pi, most of them don’t really, they converge on a fraction or multiple of Pi.)
I prefer to say discovery, but invention and discovery are two sides of the same coin anyway. Is there any invention that can’t be described as a discovery? Positive integers are an invention that humans find useful, but if the definitions are the same, and they’re pretty fundamental, mathematical usages of them are going to follow the same rules as that invented by aliens on an alien world. And they have even been defined more than once in different places in human history. So aren’t they really a discovery?
And once you define (invent) positive integers, “(Σ (i = 1 to n) i) = ((n² + n) / 2)” is undeniably a discovery. Sure, the mathematical nomenclature that I used to describe this is an invention, like the language I’m using, but it remains a fact whether anyone knows it or not. Such mathematical formulae were discovered very much like explorers discovered new lands.
idk jack about math, but if our numbers were all base seven, would pi be irrational?
@8
π’s irrationality follows from its transcendence, that is, the fact that it is not the root of any polynomial with rational-number coefficients. This fact does not depend on the choice of base used to represent it, so yes, it would be irrational irrespective of what number base is chosen.
From #3, Snowberry,
Hilbert space and other non-Euclidian geometries spring to mind. As well as Cantor’s transfinite numbers. Which doesn’t quite prove your point because Cantor and Hilbert simply expanded the mathematical language currently being used. But, at the same time, they are examples of millenia-old legacy errors being questioned then opening up a new field of mathematics ripe for discovery. I think the first example of this may be Pythagoras’ discovery/proof of irrational numbers, oh so long ago. I can well understand the legend of Pythagoras sacrificing 100 bulls in celebration of the discovery of the proof of irrational numbers (albeit the story may have grown in the telling).
@Bébé Mélange #8:
Presumably you mention base 7 because you’re seen 22/7 used for π (Pi). That’s a commonly used approximation, but the actual value of π is not 22/7, and can’t be represented by any fraction: it really is irrational.
cool, thx for the info, and for reaffirming my lack of shame about not pursuing knowledge of these eldritch things.
In the movie “Contact” there was a scene where they reailised that the Aliens were transmitting somethingorother about π in base 8.
How?
Maybe not in “Contact” the movie, but the book has a revelation about pi in base 11. Both book and movie have messages within messages within messages, but the book goes one level deeper.
An element of Neal Stephenson’s SF novel “Anathem” is the tension between those two views of mathematics, to the point of nearly being a religious war.
Nobody mentioned Gödel yet?
Maybe Leopold Kronecker was on the right track when he wrote “God made the integers; all else is the work of man.”
I would argue that “nature” doesn’t know anything about circles. The concepts of “circle”, “circumference”, “diameter”, “division”, and “number” are all abstractions we use to create mental models of some aspect of reality, but they do not exist in reality itself.
Take a sheet of paper, a 2B pencil, and a pair of compasses, and draw a circle. Now, put your sheet of paper into a scanning tunnelling electron microscope and map the entire object in 3 dimensions at atomic resolution. Take your 3D electron micrograph and blow it up so that each individual atom is around the size of a beach ball. Now, where is the circle? What are its circumference and diameter? Can you even reliably distinguish between the carbon atoms which came from the pencil and those which are part of the paper, or is it all just a fuzzy mess of varying distributions of different kinds of atoms? Even if you can, is that distinction something which is actually inherent in reality, or is it just a result of you making an enormous effort to apply your macroscopic pre-conceptions in a realm where they are neither appropriate or useful?
Looking closer at the reality we have just modelled, what is an “individual atom”, and does it even make sense to say it has a defined size?
Imagine some intelligence that was somehow able to perceive the entire comsos at once, at the quantum level. Would it divide that cosmos into discrete objects at all? What use would it have for the concept of a circle? Is “the quantum level” as we currently talk about it even a thing that really exists, or is it just another mental abstraction which models some other, more fundamental aspect of reality?
What we generally think of as “reality” is a mental model, and the nature of that model is contingent upon the senses we use to perceive some limited aspects of actual reality, and the abstractions we have created to categorise those perceptions. If we perceived differently, or thought differently, then “reality” might look very different to us.