Mathematics - Invented or Discovered

[Royce]

I've been reading Roger Penrose's " The Emperor's New Mind." He brought up a question that I once thought I knew the answer. "Is Mathematics invented or discovered?" I.E. Is Mathematics a purely mental construct or does(do) Mathematics exist, as in Plato's forms, somewhere, as Truths of Reality that we discover rather than invent?

And, by extension is Logic invented or discovered truths of the universe?

Think about it. Then give us your thoughts on the matter. It is not really as simple or straight forward as it first appears.

 

[selfAdjoint]

I have posted my beliefs before, but I'll recap here. I believe mathematical objects and relationships are all ideas in human minds. Pairs of objects exist in nature, but the number two is a human idea. Likewise objects may exist in three different places, but a triangle does not exist until a human notices it.

These mathematical ideas have a property called being well-defined. That means they can be completely understood, and people equally informed about them cannot disagree about their nature. There can be vigorous disagreements about metamathematical subjects - constructivism, formalism, whatever. But not about the nature of the mathematical ideas, This well-defined character is the reason they can be mistaken for things outside the mind.

 

[Royce]

That was my thoughts on the subject until I read Penrose's book. He brought up such things as Mandelbrot's Set or series, fractals, as an example of something accidentally discovered rather than constructed within our or his mind with this intent. Below is a link to just once site with the image of the set used. http://mathworld.wolfram.com/MandelbrotSet.html. This is just one example that he used.

His point and my question is do we really purely invent Math, Number Theory, Algebra, Set Theory, Calculus etc. or does the study physical reality and the world around us lead us to discover it to better describe and understand the universe as it is?
There are a lot of things in Math that were not invented knowing what the results and uses would be but found to be previously unknown and undreamed of products of existing Math. Another example of discovery is the Great Attractor in Chaos Theory.

 

[Stevo]

Read Wittgenstein's "Tractatus".

 

[selfAdjoint]

That was my thoughts on the subject until I read Penrose's book. He brought up such things as Mandelbrot's Set or series, fractals, as an example of something accidentally discovered rather than constructed within our or his mind with this intent. Below is a link to just once site with the image of the set used. http://mathworld.wolfram.com/MandelbrotSet.html. This is just one example that he used.

His point and my question is do we really purely invent Math, Number Theory, Algebra, Set Theory, Calculus etc. or does the study physical reality and the world around us lead us to discover it to better describe and understand the universe as it is?
There are a lot of things in Math that were not invented knowing what the results and uses would be but found to be previously unknown and undreamed of products of existing Math. Another example of discovery is the Great Attractor in Chaos Theory.

I know that Penrose is a Platonist and believes that mathematical structures exist in some "other reality" and are discovered rather than imagined.

But I don't see a true dichotomy between imagined and discovered. The Mandlebrot set is derived from the inverse of a simple complex number function. Its boundary separates the region of the complex plane where the inverse converges from the region where it does not converge. Now in my opinion that derivation is entirely a mental operation. It "looks" like discovery to the mathematician, but that is as I said before because of the sharp well-defined character that the function has. You work it out and plot the results (as Poincare did initially). Or to get more detail you program the steps you have worked out (mentally) on a computer and generate one of those pretty false color images. None of this requires us to imagine the set existing outside us in some world of Platonic forms.

You can teach another person how to generate the Mandlebrot set and then, having the necessary ideas in their heads, they can do it too.

 

[MaxNumbers]

This question seems to rest mainly in the definition of mathematics. If we say that mathematics is a language to describe reality, then I think it makes it all much clearer. For example, we see pairs of things in nature, so we give it the term "two" and begin to form the idea of "two" separate from nature. This is exactly what we do in language-- we know of emotions, and we give them names that begin to encompass the essence of anger, joy, fear, etc.
If mathematics is a linguistic tool, then we invent the terms, the names, but we are simply representing nature.
But, maybe this is just a cheap way out of the dilemma.

 

[sol2]

Self Adjoint,

I remember these previous discussions and how they arose


LAKOFF: This view was not empirically based, having arisen from an apriori philosophy. Nonetheless it got the field started. What was good about it was that it was precise. What was disastrous about it was that it had a hidden philosophical worldview that mascaraded as a scientific result. And if you accepted that philosophical position, all results inconsistent with that philosophy could only be seen as nonsense. To researchers trained in that tradition, cognitive science was the study of mind within that apriori philosophical position. The first generation of cognitive scientists was trained to think that way, and many textbooks still portray cognitive science in that way. Thus, first generation cognitive science is not distinct from philosophy; it comes with an apriori philosophical worldview that places substantive constraints on what a "mind" can be. Here are some of those constraints:

Concepts must be literal. If reasoning is to be characterized in terms of traditional formal logic, there can be no such thing as a metaphorical concept and no such thing as metaphorical thought.

Concepts and reasoning with concepts must be distinct from mental imagery, since imagery uses the mechanisms of vision and cannot be characterized as being the manipulation of meaningless formal symbols.

http://www.edge.org/3rd_culture/lakoff/lakoff_p3.html

 

[Janitor]

The threads having to do with this topic are among my favorite at the Physics Forum. I myself vacillate on what the answer is. I can see points on both side of the issue.

 

[Kerrie]

i say invented only because we do not have another intelligent form of life to prove they also have "discovered" math. math is necessary also for modern humanity, and as the saying goes: "Necessity is the mother of invention"

 

[Royce]

I have never been able to accept Plato's forms as literally existing somewhere in reality and I don't think that mathematics do either. What I do think, at least in reference to Penrose's ideas is that, along somewhat with MaxNumbers, is that mathematics may be an intrinsic and necessary part of nature, the universe or reality, whichever you prefer. As such we may thing that we invented math but it was by studying, observing or working with nature that we out of necessity discovered it.

I can just see in my mind two cavemen each with an apple and grunting their sound for one then putting them together and Eureka! they "discovered that they had two grunts instead of one thus coming to the astounding and brilliant conclusion that one grunt and one grunt make two grunts. Thus was mathematics born; and, as they say, the rest was history.

I cannot see however some reclusive caveman sitting in his cave and rather than contemplating where and what he was going to hunt, inventing our number system and theory.

This is why I question that mathematics was invented and believe that it was discovered as a basic and intrinsic feature of our reality.

 

[selfAdjoint]

We appear to have simple counting and arithmetic up to five or seven hardwired in our brains. This is not remarkable, many nonhuman animals do too.

 

[sol2]

How many animals do you know that can think of another reality under stringtheory

Platonic considerations are always a interesting feature, as it reveals, human kinds fondness of describing structures. So their was this battle for mankind to debate between discrete and continuity:)

I think royce, you might like the references to this "position" in terms of https://www.physicsforums.com/showthread.php?goto=newpost&t=24182 and their manifestation?

We have to come out of this cave thing. Heisenberg was fascinated with it?

And now, I said, let me show in a figure how far our nature is enlightened or unenlightened: --Behold! human beings living in a underground den, which has a mouth open towards the light and reaching all along the den; here they have been from their childhood, and have their legs and necks chained so that they cannot move, and can only see before them, being prevented by the chains from turning round their heads. Above and behind them a fire is blazing at a distance, and between the fire and the prisoners there is a raised way; and you will see, if you look, a low wall built along the way, like the screen which marionette players have in front of them, over which they show the puppets

http://www.ship.edu/~cgboeree/platoscave.html


But sure, there is more to it. http://www.superstringtheory.com/forum/metaboard/messages18/345.html

Some modern theories in the philosophy of mathematics deny the existence of foundations in the original sense. Some theories tend to focus on mathematical practice, and aim to describe and analyze the actual working of mathematicians as a social group. Others try to create a cognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the 'real world'. These theories would propose to find foundations only in human thought, not in any 'objective' outside construct. The matter remains controversial.

http://en.wikipedia.org/wiki/Foundations_of_mathematics

 

[Rader]

This is why I question that mathematics was invented and believe that it was discovered as a basic and intrinsic feature of our reality.

Can the infinite fractal curvature of space-time, caused by gravity, be invented, without the already existing mathematics? Certain innate constructs seem to me, to be, a necessity for our reality to unfold.

 

[selfAdjoint]

Aren't you talking about physics rather than mathematics? Physics can show how a given facet of nature is well described by a particular piece of mathematics. But it can never be proved (according to what a proof would mean in mathematics) that mathematics exactly describes all the properties of the world.

 

[Royce]

But it can never be proved (according to what a proof would mean in mathematics) that mathematics exactly describes all the properties of the world.

This is a good point. It is, however, also true that there are many things in mathematics that cannot be exactly described, computed, by mathematics. Much of mathematics are intuitive and based on assumptions.
Euclidean geometry is the first example that comes to mind. Rational and complex numbers are two others.

As Penrose pointed out mathematics itself in incomplete and cannot be completely proven according to Gödel's Theorem regardless of whether it is invented or discovered.

Speaking of Gödel's Theorem, how about that, was it discovered or invented?
Did Gödel know what he was doing before hand? Did he set out to formally prove what he already knew and already knew how to do it as is pretty much a requirement for invention. Very few things are invented accidently.

 

[matt grime]

What ec=vidence is there to back up the assertion that penrose is a platonist? i don't know, nut sincerely doubt any practisign mathematician is such a person, as anyone who understands the continuum hypothesis, say, will think too.

 

[selfAdjoint]

I apologize, it was Stephen Hawking who said Penrose must be a Platonist (the two men were close colleagues for many years). This was in a debate about "The Emperor's New Mind". Penrose responded that he regarded himself as a realist.

I believe this is a joke and that Penrose was using the term realist in its medieval sense; one who believes universals are real. Since mathematical axioms are universals, this reduces to being a Platonist in the case of mathematics, but possibly not in any other case.

 

[Royce]

In "The Emperor's New Mind" Penrose said that he was leaning toward the Platonic view himself in the same sense, I think that I am; i.e. Mathematics is an inherent part of the universe and we do not so much invent it as discover it for some of the most basic theorem's of the ancient Greeks are still true today and seem to be universal truths of and by themselves and not simple "mental constructs" of our minds. I have been paraphrasing much of what he said in the book and am using some of his examples.

I am not a mathematician so I hope that I am being clear enough and not mangling Penrose's ideas and writing to badly. As I said, before reading this book I more or less assumed that Mathematics and Logic were pure abstract constructs of our minds and so invented. I must admit that I am continuously amazed at how accurately these mental constructs can and do describe the physical world around us.

Can this be just a coincidence or is the universe really based on universal mathematic truths that we are still discovering?
It seems that the deeper we delve and the more complex and esoteric that Math and Logic become the more it seems to be discovery rather than invention.

We must remember that invention is a planned event with foreknowledge of what in required and what the desired result is with the stated intend of doing so. Discovery is the use of known facts to find new unknown facts and is often accidental, surprising, often creates more questions than it answers and is often serendipitous and mental leaps.

I must also admit that I have long leaned toward much of Plato's thinking and philosophy rather than that of Aristotle's but have never thought much of forms unless I was taking that idea far to literally myself.

 

[Janitor]

From the reading I have done on the classification of finite simple groups, there was historically a strong feeling of discovery. Some group theorists were able to show that if there was another sporadic finite simple group bigger than the ones already catalogued, it could only have an order (i.e. the number of elements within the group) of five gazillion two hundred and twenty nine udzillion and thirteen elements. Then with some additional work, they would find--lo and behold!--that there really did exist a group with precisely that order.

 

[honestrosewater]

How has information theory been applied to the question, if at all?

For example, Shannon's simple information/communication model:

message source -> encoder -> (noise ->) channel -> decoder -> message receiver

Message receiver- us
Decoder- intelligence
Channel & Noise- real world
Encoder- ?physical laws?
Message source- ?
Message- mathematics

or something like that- just an example. This seems like a very interesting and fruitful route to take. It has already given me a funny idea- noisy shadows Well, perhaps you had to be there.

 

[hypnagogue]

In what senses can mathematics be said to be a human invention? Clearly the formal systems we use to describe mathematical relationships is a purely human invention, a peculiar kind of language. However, for all of mathematics to be a purely human invention, I believe it would have to be the case that no mathematical relationships could be said to be pre-existing or fundamental aspects of reality itself, and this I take to be clearly false. In this sense, at least the basic phenomenon of mathematical relationship in general must have been discovered, not invented.

There is another, more abstract sense in which much of mathematics is discovery and not invention. Suppose a mathematician devises some odd-ball axioms within a specified formal system (call this conglomeration F) with no obvious relationship to physical reality. So far, it is uncontroversial that these specific axioms in this specific system are just abstract inventions. But now the mathematician does some fiddling around and derives some new theorems from F. These theorems are just a sort of restatement of information already contained within F and are thus, in a sense, equally as much inventions as F is. However, the mere act of 'unfolding' F, or making explicit information that was once implicit within it, is a sort of discovery in its own right. By way of analogy, I can haphazardly create (invent) a random sculpture out of clay, but after my initial act of creation I can still go about discovering new aspects of my sculpture by rotating it and looking at it from different angles.

Thus I think that there is a significant sense in which much of mathematics can be said to be discovery, not in the sense of discovering metaphysical Platonic forms, but rather properties of nature and properties of abstract systems. There is a dynamic interplay between the two, where discovery spurs invention and invention spurs discovery, where discovery is generally a discovery of completely abstract relationships that can sometimes be shown to exist in nature.

 

[Royce]

For once we agree, hypnagogue, but I think that you put it a little better than I. Your point that even in a pure human invention there may be unknown implication or features that are inherent in the system and later discoveredis a good one that I was trying to point out.
My, and Penrose's, main point is that it is nature and the study of nature that leads us to discover mathematics and the intricacies in the system.

 

[12345]

i wouls asume that it has been invented.

 

[Royce]

i wouls asume that it has been invented.

Why? How?
Why do you assume that it was invented?
How was it invented without discovering first the mathematical relationships in nature and then inventing a symbolic mathematical language to describe these natural relationships.
Mathematical relationships exist in nature and are observed by us. We invented the symbolic mathematical language so that we don't have to have oranges and/or apples on hand every time we want to perform a mathematical operation just like we invented algebra and the use of letters to represent numbers either as unknowns or as unspecified i.e. "x" ,A+B=C, 1,2,3...n etc.

The symbols and language of mathematics is beyond doubt an invention or convention but mathematics itself is a discovery of nature. Mathematical relationships exist in nature, the universe, with or without mankind. At least this is the position of Penrose and others including myself since reading his book.

 

[Dooga Blackrazor]

How do you invent something without first discovering how it is to be invented? I'd say it was discovered.

You discover how to invent something don't you? Therefore everything is discovered? Discovery and invention often come together in my book.

 

[selfAdjoint]

How do you invent something without first discovering how it is to be invented? I'd say it was discovered.

You discover how to invent something don't you? Therefore everything is discovered? Discovery and invention often come together in my book.

But if you discover something, it must have existed before you found it, no? Otherwise if by discovering it you made it newly exist, that would be invention wouldn't it? Did the Pythagoren Theorem or Wile's proof of Fermat's theorem exist before human beings conceived them? Where and how did they exist?

 

[hypnagogue]

Did the Pythagoren Theorem or Wile's proof of Fermat's theorem exist before human beings conceived them? Where and how did they exist?

Of course they did not exist as theorems, but surely their truth values existed. For instance, the Pythagoeran Theorem existed before its formal discovery insofar as anyone who happened to construct squares from the sides of a right triangle would find that the area of the square of the hypotenuse would equal the area of the other two squares combined. The name and formalism expressing the Pythagorean Theorem are human inventions, but the mathematical relationship it describes in nature is not-- that mathematical relationship existed before any human conceived of it.

 

[selfAdjoint]

For instance, the Pythagoeran Theorem existed before its formal discovery insofar as anyone who happened to construct squares from the sides of a right triangle would find that the area of the square of the hypotenuse would equal the area of the other two squares combined.

Precisely my point. The "truth" doesn't exist apart from human activity. The non-sensate universe has no "truth", just facts.

 

[hypnagogue]

Precisely my point. The "truth" doesn't exist apart from human activity. The non-sensate universe has no "truth", just facts.

Then humans discovered a fact in nature and named it the Pythagorean theorem. I don't see a big difference.

 

[Royce]

Then humans discovered a fact in nature and named it the Pythagorean theorem. I don't see a big difference.

This, of course, is Penrose's and now my point. The facts, mathematical relationships exist in nature, in the universe, in reality, however you may want to put it since the beginning and mankind discovers these relationships, facts. We invented the symobols and language or better adopted them by convention to discribe the "Natural" facts and relationships that we discover by observation and investigation of nature.
Given a circle, pi exists. Given a triangle, the Pythagorean theorem exists etc.

 

[Janitor]

To get science-fictional about it...

If the first twenty spacefaring alien species that we earthlings come into communicative contact with all tell us that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the sides--making allowances for differences in language and symbology of course--then it would have to be a pretty obstinate person to say that all of mathematics is the free invention of the human mind, and nothing more than that. Or so it seems to me.

 

[12345]

Why? How?
Why do you assume that it was invented?
How was it invented without discovering first the mathematical relationships in nature and then inventing a symbolic mathematical language to describe these natural relationships.
Mathematical relationships exist in nature and are observed by us. We invented the symbolic mathematical language so that we don't have to have oranges and/or apples on hand every time we want to perform a mathematical operation just like we invented algebra and the use of letters to represent numbers either as unknowns or as unspecified i.e. "x" ,A+B=C, 1,2,3...n etc.

The symbols and language of mathematics is beyond doubt an invention or convention but mathematics itself is a discovery of nature. Mathematical relationships exist in nature, the universe, with or without mankind. At least this is the position of Penrose and others including myself since reading his book.

exactly...i should be more specific about what i mean. i meant the number system, and the other mathematical terms have been invented by man. i believe that every thing in nature is relative to an equation.

 

[matt grime]

But Pythagoras's theorem is only true in Euclidean Geometry. It certainly isn't true in other geometries (which are more natural, despite what people think), and is a consequnce of the *definition* of geodesic.

 

[honestrosewater]

It seems that by some of these arguments, humans have never invented anything; since any invention (with the usual stipulations that an invention be intelligible and useful) is the result of observations. Surely the lightbulb existed in some form in nature, in however many fractions, but it required a kind of thought that differs from "mere" discovery. Discovery and invention may or may not require intent, yet that matters little here. Discovery entails only finding or happening upon something and does not include making modifications to or extractions from the thing discovered. The latter is invention.
Well, that's MHO anyway.
Happy thoughts
Rachel

 

[sol2]

Some like to watch the steering of boids:), and some like to watch the Dance of the Honey Bee:)

Heck, why not refer to John Nash again here, and the principals of negotiation? The consequence of conceptualization had its basis in mathematical interpretation. So given the gifted eye of the mathematicain, how watchful is s/he of the anomalies in nature?