What would it mean to say that mathematics was invented and how would this be different from saying mathematics was discovered?
Is this even a serious philosophical question or just a meaningless/tautological linguistic ambiguity?
"Intuitionists" believe that mathematics is just a creation of the human mind. In that sense you can argue that mathematics is invented by humans. Any mathematical object exists only in our mind and don't as such have an existence.
"Platonists", on the other hand, argue that any mathematical object exists and we can only "see" them through our mind. Hence in some sense Platonists would vote that mathematics was discovered.
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My personal point of view is that mathematicians invented the axioms and the rules of operation, the rest are discovered. Mathematicians invented the notations for writing down the concepts which are discovered within the universe of an axiom.
The concept of numbers exists, but we invent the notation that the glyph '1' and the sound /wʌn/ refers to the concept of singular object that we discovered. We invented the rules of matrix multiplication, but the consequences of the way we do matrix multiplications are discovered.
Most of the time, we deliberately invent a set of axioms that will lead us to discover a set of facts we want to be true. This is certainly true with imaginary numbers, we invented them so that we can discover the solutions to problems we previously were unable or difficult to solve.
There are things that are discovered, and things that are invented. The boundary is put at different places by different people. I put myself on the list and I believe that my position is objectively justifiable, and others are not.
Definitely discovered: finite stuff
By probablistic considerations, I am sure that nobody in the history of the Earth has ever done the following multiplication:
Then if I compute it, am I inventing it's value, or discovering the value? The meaning of the word "invent" and "discover" are a little unclear, but usually one says discover when there are certain properties: does the value have independent unique qualities that we know ahead of time (like being odd)? Is it possible to get two different answers and consider both correct? etc.
In this case, everyone would agree the value is discovered, since we actually can do the computation--- and not a single (sane) person thinks that the answer is made up nonsense, or that it wouldn't be the number of boxes in the rectangle with appropriate sides, etc.
There are many unsolved problems in this finite category, so it isn't trivial:
You can go on forever, as most interesting mathematical problems are interesting in the finite domain too.
Discovered: asymptotic computation
Consider now an arbitrary computer program, and whether it halts or does not halt. This is the problem of what are called "Pi-0-1 arithmetic sentences" in first order logic, but I prefer the entirely equivalent formulation in terms of halting computer programs, as logic jargon is less accessible than programming jargon.
Given a definite computer program P written in C (or some other Turing complete language) suitably modified to allow arbitrarily large memory. Does this program return an answer in finite time, or run forever? This includes a hefty chunk of the most famous mathematical conjectures, I list a few:
You can believe one of the two
Here is where the intuitionists stop. The famous name here is
The intuitionistic logic is developed to deal with cases where there are questions whose answer is not determined true or false, so that one cannot decide the law of excluded middle. This position leaves open the possibility that some computer programs that don't halt are just too hard to prove halt, and there is no mechanism for doing so.
While intuitionism is useful for situations of imperfect knowledge (like us, always), this is not the place where most mathematicians stop. There is a firm belief that the questions at this level are either true or false, we just don't know which. I agree with this position, but I don't think it is trivial to argue against the intutionist perspective.
Most believe discovered: Arithmetic heirarchy
There are questions in mathematics which cannot be phrases as the non-halting of a computer program, at least not without modification of the concept of "program". These include
To check these questions, you need to run through cases, where at each point you have to check where a computer program halts. This means you need to know infinitely many programs halt. For example, to know there are infinitely many twin primes, you need to show that the program that looks for twin primes starting at each found pair will halt on the next found pair. For the transcendence question, you have to run through all polynomials, calculate the roots, and show that eventually they are different from e+pi.
These questions are at the next level of the arithmetic heirarchy. Their computational formulation is again more intuitive--- they correspond to the halting problem for a computer which has access to the solution of the ordinary halting problem.
You can go up the arithmetic hierarchy, and the sentences which express the conjectures on the arithmetic hierarchy at any finite level are those of Peano Arithmetic.
There are those who believe that Peano Arithemtic is the proper foundations, and these arithemtically minded people will stop at the end of the arithemtic hierarchy. I suppose one could place Kronecker here:
To assume that the sentences on the arithmetic hierarchy are absolute, but no others, is a possible position. If you include axioms of induction on these statements, you get the theory of Peano Arithmetic, which has an ordinal complexity which is completely understood since Gentzen, and it is described by the ordinal epsilon-naught. Epsilon-naught is very concrete, but I have seen recent arguments that it might not be well founded! This is completely ridiculous to anyone who knows epsilon-naught, and the idea might strike future generations as equally silly as the idea that the number of sand grains in a sphere the size of Earth's orbit is infinite--- an idea explicitly refuted in "The Sand Reckoner" by Archimedes.
Most believe discovered: Hyperarithmetic heirarchy
The hyperarithmetic hierarchy is often phrased in terms of second order arithmetic, but I prefer to state it computationally.
Suppose I give you all the solution to the halting problem at all the levels of the arithmetic hierarchy, and you concatenate them into one infinite CD-ROM which contains the solution to all of these simultaneously. Than the halting problem with this CD-ROM (the complete arithmetic-hierarchy halting oracle) defines a new halting problem--- the omega-th jump of 0 in recursion theory jargon, or just the omega-oracle.
You can iterate the oracles up the ordinal list, and produce ever more complex halting problems. You might believe this is meaningful for any ordinals which produce a tape.
There are various stopping points along the hyperarithmetic hierarchy, which are usually labelled by their second-order arithemtic version (which I don't know how to translate). These positions are not natural stopping points for anybody.
Church Kleene ordinal
I am here. Everything less than this, I accept, everything beyond this, I consider objectively invented. The reason is that the Church-Kleene ordinal is the limit of all countable computable ordinals. This is the position of the computational foundations, and it was essentially the position of the Soviet school. People I would put here include
In the case of Paul Cohen, I am not sure. The ordinals below Church Kleene are all those that we can definitely represent on a computer, and work with, and any higher conception is suspect.
First uncountable ordinal
If you make an axiomatic set theory with power set, you can define the union of all countable ordinals, and this is the first uncountable ordinal. Some people stop here, rejecting uncountable sets, like the set of real numbers, as inventions.
This is a very similar position to mine, held by people at the turn of the 20th century, who accepted countable infinity, but not uncountable infinity. Those who were here include many famous mathematicians
Skolem's theorem was an attempt to convince mathematicians that mathematics was countable.
I should point out that the Church Kleene ordinal was not defined until the 1940s, so this was the closest position to the computational one available in the early half of the 20th century.
Most practically minded mathematicians stop here. They become wary of constructions like the set of all functions on the real line, since these spaces are too large for intuition to comfortably handle. There is no formal foundation school that stops at the continuum, it is just a place where people stop being comfortable in absoluteness of mathematical truth.
The continuum has questions which are known to be undecidable by methods which are persuasive that it is a vagueness in the set concept at this point, not in the axiom system.
First Inaccessible Cardinal
This place is where most Platonists stop. Everything below this is described by ZFC. I think the most famous person here is:
I assume this is his platonic universe, since he say so explicitly in an intro to one of his more famous early papers. He might have changed his mind since.
Infinitely many Woodin Cardinals
This is the place where people who like projective determinacy stop.
It is likely that determinacy advocates believe in the consistency of determinacy, and this gives them evidence for consistency of Woodin Cardinals (although their argument is somewhat theological sounding without the proper computational justification in terms of an impossibly sophisticated countable computable ordinal which serves as the proof theory for this)
Possibly invented: Rank-into-Rank axioms
I copied this from the Wikipedia page, these are the largest large cardinals mathematicians have considered to date. This is probably where most logicians stop, but they are wary of possible contradiction.
These axioms are reflection axioms, they make the set-theoretic model self-simialar in complicated ways at large places. The structure of the models is enormously rich, and I have no intuition at all, as I barely know the definition (I just read it on Wiki).
Invented: Reinhard Cardinal
This is the limit of nearly all practicing mathematicians, since these have been shown to be inconsistent, at least using the axiom of choice. Since most of the structure of set theory is made very elegant with choice, and the anti-choice arguments are not usually related to the Godel-style large-cardinal assumptions, people assume Reinhardt Cardinals are inconsistent.
I assume that nearly all working mathematicians consider Reinhardt Cardinals as imaginary entities, that they are invention, and an inconsistent invention at that.
Definitely invented: Set of all sets
This level is the highest of all, in the traditional ordering, and this is where people started at the end of the 19th century. The intuitive set
These ideas were shown to be inconsistent by Cantor, using a simple argument (consider the ordinal limit plus one, or the power set of the set of all sets). The paradoxes were popularized and sharpened by Russell, then resolved by Whitehead and Russell, Hilbert, Godel, and Zermelo, using axiomatic approaches that denied this object.
Everyone agrees this stuff is invented.
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This is only a partial answer:
As a mathematician, I have been asked this sort of question from time to time. Like most other mathematicians, I tend to sort of evade the question, because it's tricky. Usually, the question is put in the form, "Are you a platonist?"
The reference here is to Plato's eternal form that we are able to recognize, and that allows us to recognize the world around us (it is not obvious, afterall, that we should still be able to recognize an amputee as a human when we first see him or her, for example). When forced to continue, I usually respond "No."
I think the fundamental problem with Platonism is summed up in Brian Davies's paper, aptly titled "Let Platonism Die." I also add - if a mathematical 'discovery' hasn't yet been discovered, does it exist? A Platonist would say absolutely. An intuitionist would either say that it does not exist, or it exists only in the sense that some current or future mathematical system, devised and formulated vulgarly by humans, will lead to many more theorems - i.e. it exists only as an extension of what we have already created.
But ultimately, I don't think that this distinction is very important aside from the theistic or neural implications. A Platonist would say that when we recognize a triangle, for example, it is because we are recognizing the Form of a Triangle, some idealized, perfect, transcendental object. This makes a lot of sense, because Platonism obviously has at its roots Plato, who read much into the divine relationship between mathematics and the world espoused by Pythagoras.
As a final note, I should say that many well-known mathematicians lie on both sides of the fence. The most famous Platonist, I believe, is Roger Penrose, who is most famous for his creation of dozens of non-obvious tessellations and tilings.
I think the words "invention" and "discovery" are a bit poor to describe the birth of mathematic if there is one. It makes no sense to me to say mathematic has appeared as when Christophe Colomb discovered America or was invented as the boomerang.
The word mathematics might have been invented, the language in which the mathematics are written might have been invented but the abstraction movement from the real word, the structured synthesis that it undertakes, all that give thickness to mathematics themselves (it depends what you call mathematics) are part of mankind. You don't ask if beauty has been discovered or invented ?
My personnal point of view is that the question "what is mathematics" would be more serious, I would found even more interesting "why do we do mathematics".
Formal mathematics is created by people, and doesn't necessarily relate to anything in our world.
However, the history and progress of mathematics is many times related to applied mathematics, which is related to our physical world.
or in other words - geometry will remain valid even if we will find out that it doesn't hold true for our physical world (and actually it doesn't...) - but it's hard to believe many people would have started researching this field as a pure abstract field, with no relevance to real problems of construction, navigation, etc.
Mathematics is an abstraction. As such it is invented by humans to deal with concrete things is a more practical manner, by giving us generic tools to deal with the specific.
Later more mathematics was invented to deal with the abstractions of earlier maths, leading to more and more complex abstractions, but the invention of math was done to deal with concrete things, like geometry and trade.
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Mathematics is a lot of things: there are basic/complex entities/structures, proof strategies, algorithms, formal manipulations... in order to try to answer your question I think we should make some distinctions between different matematical entities/activities where the "creative" part of the thought is more or less relevant. Moreover some parts of mathematics seems to be neither discovered nor created, they seem to be just "given" embedded in our natural language grammar.
Some examples of math entities/activities that:
I'm going to posit, admittedly without any research whatsoever about those who've preceded these thoughts, that an "invention" is a kind of "discovery," and that whether a thing qualifies as an invention is—yup, you saw it coming—subjective.
For example, we might say that the wheel was "invented" on grounds of (1) non-naturality (originality), and (2) intention. That is, prior to the wheel, circle-and-axle forms did not exist in nature, and so of course no one could apply it to the facilitation of movement. Furthermore, it's hard(er) to imagine someone carving a circle with a hole, then carving a spoke, then putting the two together, without having the intention of rolling the circle on the spoke, in mind. These circumstances give us cause to say that the wheel was "invented."
But, it's not impossible to imagine, either, that someone might have carved a circle with a hole for absolutely no reason to do with the concept of rolling, then happened to stick a stick in the hole (again, for no premeditated or relevant reason), and only then (or sometime later) realized its property of rolling. Note how in this case, we're more inclined to call the wheel a "discovery!"
I think we tend to call novel discoveries with premeditated results, "inventions."
So, I would say mathematics, as a general notational/deductive system, was mostly invented. But its concepts were discovered. (And even some notations were indeed discovered, while striving for convenience, concision, and pictorialization!)
First, Quine: "..[If externally true] the definitions [of mathematical laws] would generate all the concepts from clear and distinct ideas, and the proofs would generate all the theorems from self-evident truths." "...the truths of logic are all obvious or at least potentially obvious..[but] mathematics reduces only to set theory and not to logic proper." -Epistemology Naturalized; Chapter 39.
The implications are bleak for the ontological objectivity of mathematics. For a fact to reduce to certainty one must present sensory evidence (to be "self evident"). Consider, I see that things fall to earth and stay there. I explain this to myself with physics. What I see is not physics. Physics is a framework invented to generalize what I am perceiving.
A 1 and a 1 on a sheet of paper are not the same as a 2 on a sheet of paper. 1 is the smallest prime#, for example, while 2 is the smallest even prime, among myriad other differences.
An apple on a table and an apple on a table is not the same as two apples on a table, as the set of two apples could be different apples. I cannot cube two apples, except to make pie. But I cannot make pi with an apple.
The value of a dollar is measured mathematically. But if humans disappear, the piece of paper remains, while the value disappears with humans. Things stick to the earth regardless of our existence, but the theory describing our perception of gravity does not.
The epistemic objectivity of Mathematics is ontologically subjective. It exists only in our minds. Something that exists only in our minds can only have come into existence within our minds. Something that does that is invented.
I think it's hard to say. If you believe that mathematics has been discovered, you must assume that "something" is out there, something we can interact with, of which we have been unable to prove existence so far.
However, even assuming that there are ideas out there, I believe that there is no reason to think that humans should be, in any way, able to understand them. As David Deutsch famously said, the fact that we understand the laws of Nature, is pretty much like saying that you land on another planet, and find aliens completely able to speak to you in english.
Last but not least, it is possible that our models of how the Universe works are completely wrong. Hence, we are talking about ideas derived from our models that may be, ultimately, way off the truth.
My view on it is that Mathematics is a system invented by humans to represent things we otherwise can or cannot perceive. For example, we can perceive an object through vision and know it's a triangle, however, our vision alone does not tell us the length of the legs of the triangle. We need math to represent that for us.
JUst to further my point, consider Calculus. Two people who were on completely different sides of Europe, Leibniz and Newton, created a system that that both do the same thing. For Newton, f'(x) is the same as Leibniz' df/dx. Both of them yield a function that represents the slope at any given point on the original function, f(x). They invented a system to represent something we otherwise couldn't perceive (which was pre existing - The shape of a mountain should be enough to prove that the slope exists naturally), the only difference was their notation.
Mathematics is normative. That is clear when one reads Euclid and Lobachevsky in juxtaposition, or Euclid and Descartes, or Euclid and Leibniz or Newton, or Leibniz and Newton and Dedekind, or Dedekind and Canton, or Canton and Godel, etc., etc.. Geometry is clearly normative, as we have different geometries (although one might claim, "yes, but they can all be transformed into one another"). But the argument goes like this: there is no other arithmetic; and thus, in counting (and its extensions), we are discovering something fundamental to the universe. Of course, such an answer supposes that Euclid and Dedekind are talking about the same arithmetic. Is that even possible? No. There's no room, in Euclid's conception of number (think of Books V and VI of the Elements), for Dedekind's cuts, and thus, no room for a whole host of numbers that are incompatible with Euclid's concept of number. And if you think that the concept of number is fundamental to a conception of arithmetic, then it would seem that every time we "add" new "sorts" of numbers (which are invented by new sorts of functions), we create a new arithmetic. But, someone might say, "that's all well and good, but we really just subsume those other arithmetics under what we call arithmetic--there's really just one arithmetic." But that would be like saying "wave-mechanics really just subsumed ordinary mechanics...." Such a statement doesn't make any sense.
In line with many others' probing at just what 'invented' means, invention and discovery can be viewed as the same thing, as both require the application of a set of steps along with various objects under consideration. Even when discovering, say, a continent, the notions of continent-ness and America-ness are both inventions, nonetheless. And even when inventing, say, the internal combustion engine, the laws of physics which allowed such a device to exist were in place before the invention, and thus the particular arrangement of parts which effects its existence was discovered.
If by "was it discovered?" you mean "was it there all along?," I think the answer is "yes." Consider that we can use math to "predict" the past ("retrodiction"). A similar concept is "hindcasting," where the validity of a scientific model is tested against data that was recorded before the model was even invented. Presumably, in order for retrodiction/hindcasting to work, the mathematics had to be there all along, constraining the evolution of the universe. If you buy this argument, this suggests that mathematics was there all along, or "discovered."
Of course, other definitions are possible.
The Black-Scholes equation describes the price of a stock option over time. Since the concept of stock options, financial markets et cetera were invented, not discovered by humans, does that suffice as an arguement that mathematics was invented? If there was no such thing as a stock option, there almost certainly wont be the black-scholes equation. The black-scholes equation would never be out there waiting for us to discover it if there was no such things as a stock option.
If one claims that although a stock option was invented, the black-scholes equation can be said to be discovered, how many more mathematical theorems, equations, models and so forth are out there that are waiting to be discovered, dependent on our future "inventions and creations"?