Was mathematics invented or discovered?

Question

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? In the context of mathematics, could “invented” and “discovered” mean the same thing?

Here is a headline from 2008 which says, Huge new prime number discovered, which is an example of a human intuitively feeling that at least some mathematical truth is a matter of discovery.

Just to make it clear:

• discovery = finding something that existed before, for the first time (e.g., a frog, black holes)
• invention = intellectual creation of something that did not exist before (e.g. a system, concept, pattern, idea, object, or information)

Answer 1

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.

Answer 2

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 probabilistic considerations, I am sure that nobody in the history of the Earth has ever done the following multiplication:

9306781264114085423 x 39204667242145673 = ?

Then if I compute it, am I inventing its 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:

• Is chess won for white, won for black, or a draw, in perfect play?
• What are the longest possible Piraha sentences with no proper names?
• What is the length of the shortest proof in ZF of the Prime Number Theorem? Approximately?
• What is the list of 50 crossing knots?

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:

• The Riemann hypothesis (in suitable formulation)
• The Goldbach conjecture.
• The Odd perfect number conjecture
• Diophantine equations (like Fermat's last theorem)
• consistency of ZF (or any other first order set of axioms)
• Kneser-Poulsen conjecture on sphere-rearrangement

You can believe one of the two

• "Does P halt" is absolutely meaningful, so that one can know that it is true or false without knowing which.
• "Does P halt" only becomes meaningful upon the halting of P, or a proof that it doesn't halt in a suitable formal system, so that it is useful to introduce a category of "unknown" for this question, and the "unknown" category might not eventually become empty, as it does in the finite problem case.

Here is where the intuitionists stop. The famous name here is

• L.E.J. Brouwer

Intuitionistic logic is developed to deal with cases where there are questions whose answer is not determined as 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 intuitionist perspective.

Most believe discovered: Arithmetic hierarchy

There are questions in mathematics which cannot be phrased as the non-halting of a computer program, at least not without modification of the concept of "program". These include

• The twin prime conjecture
• The transcendence of e+pi.

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 hierarchy. 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 Arithmetic is the proper foundation, and these arithmetically minded people will stop at the end of the arithmetic hierarchy. I suppose one could place Kronecker here:

• Leopold Kronecker: "God created the natural numbers, all else is the work of man."

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 hierarchy

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 levels of the arithmetic hierarchy, and you concatenate them into one infinite CD-ROM which contains the solution to all of these simultaneously. Then 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 arithmetic 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

• Yuri Manin
• Paul Cohen

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

• Thorvald Skolem

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.

Continuum

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 the 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:

• Saharon Shelah

I assume this is his platonic universe, since he says 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 the 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)

This includes

• Hugh Woodin

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-similar 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 inventions, and inconsistent 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

• The set of all sets
• The ordinal limit of all ordinals

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.

Answer 3

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.

Answer 4

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!)

Answer 5

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".

Answer 6

Both.

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.

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.

Answer 7

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.

Answer 8

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.

Answer 9

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:

• seem embedded in our grammar: classical logical operators, classical deduction rules, tautologies, natural numbers
• seem more discovered: non-trivial general fact in a given structure (ex. fermat's last theorem), finding general patterns, classifications, finding counterexamples
• seem more invented: definition of new non-trivial structures (ex. complex numbers, quaternions), finding new non-trivial proof strategies.

Answer 10

This is a serious question and it is the same as saying: is the knowledge in mathematics universal or a human construct?

Pi (the number, regardless of its base) and many other things are universals, mathematics are discovered to that extent. Then they can be used to formalise inventions that may prove to be wrong, right or paradoxical, in the same way the knowledge (discovered) about horses and rhinos can be used to (invent and) speak about unicorns (that were never discovered).

Can we say (as many answers point here) that biology was invented because of unicorns?

Answer 11

I think the distinction between discovered and invented is mostly about how one chooses to define these words. My personal definition would be that when you can reasonably assume that many other people can in principle find the same thing X, then X can reasonably be said to be discovered, but when X is pretty arbitrary, like a particular notation, then it's invented. For example, different people can discover the Mandelbrot set, and various relationships and figures in there:

In the above image the colors are an invention, not a discovery. Different people will maybe choose similar coloring here, but I think it's pretty much an artistic choice. The colors roughly reflect how fast a point in the complex plane will head off to infinity under a certain repeated square-and-add operation, but they depend on a lot of parameters (including how many iterations one deems sufficient to establish the wayward nature of a point), including, of course, some particular color palette.

I think this illustrates nicely that the very same mathematical beast can have aspects that are discovered, and aspects that are invented. ;-)

Answer 12

Math is a system made up to quantify, measure, understand, and determine things by mathematical proof, logic, analytical reasoning, and common understandings. It's also an abstraction as well since the actual theoretical basis given on the implementation of the something will usually differ in practice atomically, etc.

Math is an endless study of conjectures that is agreed upon by people subscribed to such a phenomenon. Math has been used for centuries to keep track of things, measure qualities of things, and in modern days to analyze and interpret highly complex conjectures, theories, and explanations of everything around us.

Was it invented or discovered? Philosophically speaking, is anything ever really measured or discovered?

Somethings just are, and to our best knowledge we have a system, math, to quantify and analyze things.

Math never "was" anything until it was agreed upon, brought in to use, and implemented, agreed on, and understood. Such highly complex systems never were used by the biological creatures way before us, e.g. fish, bacteria. Quantity is just mass without numbers, and quality is just coincidence without observation.

A response to another question I found here that spiked my interest:

why does the concept of number exist, but the concept of complex number invented?

The concept of everything tangible and/or intangible only exists to be understood based on the reality and observation of the phenomenon around it, how those phenomena perceive it, agrees upon understanding it, and how well that system can accurately model the underlying reality of it. To a human a ball is something you kick, throw, catch, has shape, mass, volume; to a dog it's something in its way. The reality is that if there's a reality under the underlying concepts we try to figure, only such a system invented will try to emulate the process of understanding it more and more.

The question also touches on the grounds of everything around us, and its totality. Let me give you an idea of why I am proposing that math is an invention:

Before people could even count, or even existed there were always many different biological structures, masses, gases, inanimate objects, and collective existences outside of a singular model, single perception of electromagnetic radiation's visible light, eyeballs, brains, or classification itself. Before we evolved did dinosaurs, assuming you believe they existed, count and classify the world around them? Probably to a mere, limited extent, but not anywhere near how most people would think of it. All biological creatures that have evolved past the bacteria have gained perception, analyical minds, and the capability of complex thinking to become better suited to the existence to around them. None of them ever came anywhere close to modern humans.

I doubt the fish in the sea can accurately model multiple perceptions of visible light on masses, and use their brains to visualize this as two separate objects, thus, manipulating the abstraction of items, beings, or existences around them. However, we look at two things and agree these are two things. We see two rubber balls on the floor, and we come to the immediate conclusion that they are two distinct objects. But are they really two things, or have you just subscribed to a common method to segregate objects based on human evolved, educated, or brain limited rules?

Point is, you see two non-connective items, and you classify/label them as two. You aren't, in most cases, visualizing the ball as a synthetic base of polymers, isoprene, and other chemical elements and masses that constitute its existence within electromagnetic radiation in an atmosphere. Therefore, you have classified the existence of two balls based on segregating instances of light, however, you are only using a system to do so that is 100% limited to your brain's understanding.

Without a system, understanding, or method of perception everything would exist, but would not be calculated, observed, or manipulated.

Answer 13

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.

Answer 14

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"?

Answer 15

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.

Answer 16

From a Neo-Intuitionist perspective, to the degree mathematics is invented, it is still discovered.

Did we invent, or discover the consonant 't'? We discovered that our mouths reasonably make that sound, across a wide swath of our species. But we decided that this was an important thing, and in so doing, we invented the idea of 't'. We invented a consonant by discovering a fact about ourselves.

From this perspective, mathematics is a set of ideas to which humans are naturally attracted in a given way. But those ideas themselves are a product of the human mind, the way the consonant 't' is a natural product of the human vocal apparatus. Those ideas arise out of individual humans, who can be considered to invent them. (Someone first uttered the sound of t. Someone first asked if -1 has a square root, or whether infinity comes in various sizes.)

But mathematics chooses out the ones that feel a given way and isolates those that appeal broadly to a given emotional reaction. In that sense it is a branch of psychology which discovers things about human thought.

It elaborates those ideas to a degree that makes it seem like it is creating things, but really, it is exploring our shared fund of ideas for ones that seem purely symbolic and not worthy of questioning, and sees how their consequences fit together.

Answer 17

If only we would get the question right, we may be able to get the right answer. The problem is, is invention discovery or creation? As a seven times patented inventor, I will tell you that invention is, at least to a great extent, discovery. As my patent agent explained, what is invented is a "method", a way of getting a job done. During the process of invention, one tries on a gazillion methods of getting the job done that don't work. When one does discover a method that does work, well, one has an invention.

The proof of discovery verses creation, is the proof of reproduction. When a person who has never seen a wheel before tries to solve the problem of causing heavy objects to move, he may very well re-invent the wheel. This happens all of the time with inventions. One comes up with a method of solving a problem, only to discover that someone else has patented that invention before him. Creativity is not like this at all. If two people truly independently come up with the same creative product, then their creative product is, well, simple. In fact programs are used to analyze college papers for plagiarization. They seek matches in a 7 word sequence because it is unlikely that two people independently come up with seven little words strung together the same way.

So let the question be, "is mathematics discovery or creation?" Ask the anthropologist to seek out the mathematical methods of other cultures. Surely these methods would be extreme subsets of our math. However, they still have some simple consistencies. Two plus two (though represented with different words) equals four. The fact that two cultures independently come up with the same logic sets establishes that mathematics is discovery, not creation.

Answer 18

My elementary math lecturer likes to say

God created the number 0, and the successor. The rest was invented by mankind.

I think there is some truth in this quote, even if you don't believe in God. So to answer your question: I'd say that the very basis of math was discovered, but most of the sophisticated math was invented.

Answer 19

Let us consider that the quest to settle the question as one or the other is a false dilemma of sorts, one in which, literally speaking, it is neither discovered (like a lake is discovered) nor invented (like a lightbulb). Rather it is constructed. Taking such an approach let's us concede that the application of the verbs 'discover' and 'invent' are, strictly speaking, metaphors derived from physical experience, and not literal language, applicable to the abstract domain of thought. This of course is roughly the position of Luitzen Egbertus Jan Brouwer which is entirely underappreciated by mathematicians.

Thus, to understand this position let us note two passages, the first in the SEP's Intuitionism in the Philosophy of Mathematics:

Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician L.E.J. Brouwer (1881–1966). Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds.

And the second comes from the SEP's article on Constructive Mathematics (SEP):

Constructive mathematics is distinguished from its traditional counterpart, classical mathematics, by the strict interpretation of the phrase “there exists” as “we can construct”. In order to work constructively, we need to re-interpret not only the existential quantifier but all the logical connectives and quantifiers as instructions on how to construct a proof of the statement involving these logical expressions.

Hence, like other long-ranging debates (nature vs. nurture, mathematical realism vs. nominalism, objectivity vs. subjectivity), we find a third position that takes the best of the two others in the thread-worn debate.

Answer 20

This is just an exoskeleton of an attempt to write an answer that I think no one else has written here yet. Work in progress.

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Problem 1: What do ‘invented’ and ‘discovered’ mean?

One can freely consider what strikes them as the most prominent difference between these words. For example, it feels like “invented” is the creation of something that did not exist before, and “discovered” is the finding of something that did. Put more abstractly, we can make use of the concept/category of existence or to exist: it appears one of the most critical distinguishing criteria here is, did the thing exist or not, prior to the significant moment. What significant moment? Well, the general thing that can be either discovery or invention. What is that? It appears the second component in an equally abstract form is: the moment it became known (where I suppose ‘known’ has many possible variants, like apprehended, encountered, etc.)

I like this analysis so far - it doesn’t help us, but it completely changes the terms of the question. The question is about metaphysics, and it is about epistemology: to what extent did something exist or not, prior to it becoming known?

Yet, I believe both “exist” and “know” are surprisingly multifaceted topics in their own right (for example, see the Gettier problem) and Existence (SEP).

First, we can try to establish some terms of what existence and knowing are, and see how they may apply to mathematics. (But after, I would like to come back and ask, How do I know that my understanding of invention and discovery is correct? Is there a way to prove that one has found the correct definition?)

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Existence is not a topic I have thought much about, but I have frequently felt like the term refers to many different things which are not often discriminated from each other. Especially when ever anyone claims something does not exist, it sneakily affects our belief about some thing, whereas it should cause us to ask what that assertion tells us about what the person saying it thinks “exist” means, in that context.

I’d like to suggest a fundamental paradox of existence, for starters… anything that can be (meaningfully) talked about (to use informal language that is meant to mean something specific), exists on some most primary level of existence possible. Unicorns are not present insofar as we know, in our world; yet the absolute lowest level of “existence” must be able to accommodate unicorns. I claim this because, if unicorns do not exist, how do you describe whatever it is they do do? You might find yourself forced to say, “it’s an idea”. Are we then committed to the thesis, “the idea of a unicorn exists, but unicorns themself do not exist”. I think we could still argue that there is something called “existing in conceptual space”, in contrast with a hypothetical idea that does not exist. (After writing this, I am already considering refuting this… we become confused by language when we insist unicorns have some kind of conceptual existence… it’s almost like muscle memory, we can talk about their properties as if they exist, but that does not necessarily require them to “exist in conceptual space”… perhaps they are only an idea after all - perhaps this is a “physicalist” view, where even ideas are ultimately certain fluctuations of electrons in space. More on this later.) It is impossible for me to give an example of an idea which does not exist in conceptual space: if I could, it would exist in hypothetical conceptual space. Thus, it is unnameable. It is the difference between the word/concept “nothing”, and nothingness itself: the idea of nothing is a thing; but nothing itself is not, or that would be a contradiction.

I am rambling because this is the first time I am considering these ideas. Maybe the temporary conclusion for now is, similar to the idea of “modal bases”, we will try to never claim that something does or does not “exist” as a general property, but we will always specify a type, or flavor, of existence; a kind of predicate: “it is physically present in this world, at this time”; “it is a mental phenomenon that I can summon into ‘existence’ by mentally thinking about it, at any time, at will”; “it is something which I know will be the case, from reason”; “it is not currently present in my consciousness / perception”; etc.

To wrap this up for now, I will not worry too much about the definition of “know”. Let us say that when an idea - logical or otherwise - is present in a mind, this is enough to say it has been known: it has been encountered, ‘made contact’.

A third part of the equation is, what is mathematics? Again, I have more to say on this, but I have been most influenced by the perspective that logical truth is necessarily necessary. It would be incoherent and impossible to live in a world where the laws of math as we know them do not hold. The laws of math are not contingent, but are an inevitable manifestation of the only way anything could ever be. (Or so I claim / hypothesize).

Now we have drawn the terms of the question, a little bit:

Do ‘necessarily necessary’ propositions exist, prior to being encountered by a mind?

I feel like the question looks a little simpler now. Because at least we reduced it to only one parameter: what do you mean by ‘exist’?

If you mean, “a structural manifestation of that proposition had phenomenological existence in space time from the point of reference of your consciousness”, then by definition, the answer is no, but this is trivial: it is sort of like solipsism combined with idealism. If you think “truths” aren’t really there until somebody thinks them, and if you think nothing is really there until the candle of consciousness of a human mind bestows phenomenological reality on them, (sort of like idealism), then it is easy to say that that mathematical truth did not really exist, until you thought of it.

We might say, “but wasn’t it logically predetermined to be that result”? We could say, “Yes, but now that we have such a rigorous definition of ‘existence’, that isn’t a problem: the fact that logic determines that truth is not the same thing as that it as of yet exists.”

So, I offer this as a starting basis for further development on this topic.

Answer 21

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.

Answer 22

Mathematics started as a discovery. It was the discovery of inequality. It was the discernment of more suffering and less suffering. And since then we have invented many things like unity , numbers , geometry, trigonometry, algebra etc to solve our problem of inequality. Inventions created new fields of research and new discoveries were found embedded in it.

Answer 23

I consider an answer too simple if it just affirms one of the alternatives and negates the other.

Naming just a few eminent contributions to mathematics: Complex numbers, set theory, theory of schemes. E.g., the concept of a set has been invented by Cantor, it did not exists before. After the basic concepts like set, power set, cardinality etc. had been invented, the Continuum Problem was discovered, hidden deep in these concepts.

Therefore I compare mathematics to a game like chess: Inventing new mathematical concepts is like creating new rules of the game. Playing a match means to discover the consequences of the rules and to solve the problems posed by the rules.

My conclusion: The rules of the game of mathematics have been invented. Following the rules mathematicians then discover some challenging matches.

Answer 24

Every mathematician can only discover mathematics.

Yet, mathematics is an invention.

And this is no contradiction.

Mathematics is fundamentally dependent on the human mind, and more particularly on human deductive logic and on the human perception of the real world, so it is a sort of invention of the homo sapiens species, not one of individual mathematicians. Human mathematicians are unable to invent anything logical which does not follow logically from human nature.

Thus, every mathematician can only discover what is already there, implicit in human nature, and since the mathematicians we know of are all humans, they will discover or rediscover the same things.

This is why mathematicians come to believe that it is a given, hence the Platonist view.

The Platonist view is wrong because while mathematics is given to every mathematician, it is not given to the human species. It comes with, or is part of, its own nature, so to speak. Human mathematics does not exist outside the human mind.

The human species is itself implicit in nature, so human mathematics is implicit in nature, but it is set out according to human logic and human perception of the real world. So, at best, the Platonist world is nature itself. This is clearly not what mathematicians mean by "Platonist", but this is the only reasonable option.

Answer 25

Not All Math is Equally Created

Some invented. Some discovered. Some sought. Some built. Some inspired. Some devised. Some engineered. Some gets doodled into being.

Math gets published. Reproduced. Grows. Adapts. Gets developed. Gets outdated and obsolete and forgotten. Evolves.

Is there benefit or need or rule or reason to create two boxes labelled...

• "INVENTED"
• "DISCOVERED"

... and then try to stuff all mathematics into one box or the other box.

The boxes are inventions.

Answer 26

This is an observation I can't remember where I heard, so would be greatly obliged if anyone else knows. But I think it's a killer line of argument.

Consider that somewhere in the set of all rational numbers, is the answer to any question you could ask (taking the numbers as e.g. ASCII codes). Yet knowing this does not give you these answers. It would take enumeration of a number, and then a relational process to check it and confirm it is correct.

So by this model, enumeration, and checking relations, are not magically external to the properties of a number, but fundamental to it. Invented not discovered, QED.