Is a theorem in mathematics discovered or invented?

Hasan,

In terms of whether we discover some theorems or not, the term discovery has special meaning for a Platonist. That is, the discovery of a theorem would be explained in terms of our putting together relations between pre-existing forms (it is a bit like writing down our observations and conclusions that we have found outside ourselves). There is a 2010 article on Plato's Ghost in a 2010 issue of the Notices of the Amer. Math. Society. I can send you a copy, if your are interested.

Hasan Hadi Khaleel

Prof. James,

Thank you for the clarification. Yes, please send me a copy of that article to take a look at it.

Regards

Hasan

Ibrahim Eltayeb

To discover is find out something that is already there while to invent something is to make up something from a number of things put together.

Here is a copy Yuri Manin's AMS Notices article about Plato's Ghost, a review of Jeremy Gray's book: Plato's Ghost: The Modernist Transformation of Mathematics,

Princeton University Press, 2008.

Sarvesh Rawat

well i will start it by taking an example that i i read somewhere a long time ago . of a dog when it catches the ball in th air . By the help of mathematics we can predict its trajectory due to gravity and other forces and can tell where it will land. but amazingly a dog also knows where it is going to land and catches it before it touches the ground..the dog clearly estimate the landing point regardless of our mathematical equations , proofs in the 3 dimensional trajectory it exactly know the path.

does it discover math ., or invent it or neither?

you can say it discovers ..by following a iterative process of repetition it observes the patterns through the space and it discovers that it could be applied for the future events.its the same like in a cricket match a batsmen play a shot just by looking at the hand movement of the bowler he discovers where the ball will bounce in front of him ..

well another thing that we can say about is its a "tool" or mechanism to describe the happing around us..this tool includes the ways to relate between quantities ,to find patterns so that they can be put under some theorem or axioms.that were inverted by human beings as a helping tool to explain the things.

we invented mathematics to describe the nature but everything that it describe is already there in world.

its a tool which help human to describe about the nature it was invented as a tool ,but the notions of mathematics

by which it explains and calculates were discovered by humans with the evolving civilization but they were always there !!!!!!

Sarvesh, I think the example you gave has to do with physics and the laws of nature rather than mathematics. The laws of Nature are indeed discovered as they already exist and are just waiting that someone discovers them.The case of Mathematics, on the other hand, seems to me to be a quite different story.

There are cases where a work in mathematics is produced without being motivated by any application. I believe that the mathematical tool thus produced is the result of a pure invention.

On the other hand, there are other cases where a work in mathematics is rather motivated by an attempt to find a more appropriate tool to model an already discovered physical phenomenon. In this case also, I would say that the tool is invented even though the motivation for producing such a tool originated from a discovered natural phenomenon.. .

@Sarvesh Singh:

@H.E. Lethihet:

James,

I agree with you because If we look back in history, we can see that progress in mathematics has triggered that in sciences and vice-versa. Therefore, I do not see why this process would stop in the future.

As to the question of how much is yet to be discovered, I can answer with confidence that what we already know only represents a tiny part of what we will discover and that what we will ever discover only represents a tiny part of what actually exists.

Sarvesh Rawat

okay sir fine ...... lets say it is invented But still there are questions that are left for example there were so many civilizations from history completely isolated and independently from each other discovered the same mathematical principle ??

and what about other intelligent life (if they exist) outside earth...the principle of nature would be same for them in this universe..they have the same notions of the things but their representation or syntax may vary..in this regard it is discovered ...

anyway from my limited knowledge its quiet hard to distinguish it like either /or.

A theorem in mathematics is really nothing more than a collection of symbols that someone has taken the trouble to write down in a meaningful fashion. In order to 'discover' the theorem it is first necessary to 'invent' the necessary symbols, defining an associated meaning with each and the rules for combining symbols with mathematical operators. For example the symbol 2 can mean either the real integer 2 or an instruction to square something if the symbol is written as a superscript.

Thus in Plato's Cave it is the mathematical representation (aka notation) that is 'invented' whilst the underlying concepts always remain to be 'discovered'. This illustrates the importance of developing good notation in producing the theorems that constitute mathematical understanding, but the same argument could be applied equally well in a field like musical notation. P(event)=0.1 is a great notation but what does a probability of one in ten actually mean? A mediaeval musical score certainly looks interesting but what does it sound like?

A good example of notation that has developed historically is the so-called Binomial Theorem describing the expansion known as Pascal's Triangle. The reason the notation (x + y) super n = (the familiar binomial expansion) is such a brilliant invention, not just a mathematical representation that enables us to construct a Binomial Theorem for integral powers of real positive numbers, is that it enables us to think abstractly about substituting fractional and/or negative indices for n. We allow the notation to develop a life of its own, and play with it independent of its parent concept. Having crossed this Rubicon we find ourselves squarely in the realm of mathematical 'discovery'.

Historically it is worth reviewing representations of the binomial expansion e.g. in Newton's Principiae to see how cumbersome notation can impede discovery. Although it didn't seem to stop him for long!

@Richard Hibbs:A theorem in mathematics is really nothing more than a collection of symbols that someone has taken the trouble to write down in a meaningful fashion....

There are theorems that are assertions containing only words. Here are a few examples:

Theorem (Euclid) There are an infinite number of prime numbers.

Theorem (Bolzano) Every continuous function from one metric space to another preserves connected sets.

Theorem (Hoggar) A net is centered, if and only if, it admits a rhombus cell.

Proof: see S.G. Hoggar, Mathematics of Digital Images. Creation, Compression, Restoration, Recognition. Cambridge University Press, 2006, p. 60.

And widely acclaimed theorems tend to work their way into the literature as definitions.

Although I think that mathematics are invented rather than discovered, it is not possible for me to prove that I am right or that those who think otherwise are wrong. I can only attempt to explain my view.

To discover = to find or become aware of something that was previously unknown to us but that already existed before our discovery.

To invent = to create, build or design something (or a concept) that was previously unknown to us and that did not exist before our invention.

In the case of mathematics, we invent first a mathematical concept and we define the axioms. By doing so, we implicitly fix all the fundamental rules (FR) of the game and these rules will apply inside our newly designed mathematical sub-world (MSW). Once this is done, our MSW will already include all the properties that follow directly or indirectly from the FR. Usually, most of these properties were totally unknown to us at design time but were nonetheless already present when we created our MSW. Next, by exploring our MSW, some of these unknown (but existing) properties would progressively emerge and we would "discover" them (while some others will remain unknown and might never be "discovered").

Therefore, we could say that we invent a mathematical tool; then we "discover" some of its emergent properties. However, in my opinion, such a "discovery" is merely an illusion created by our inability to foresee, at design time, all the features that we have, ourselves, unknowingly placed in our invented tool.

@H.E. Lehtihet::... we implicitly fix all the fundamental rules (FR) of the game....

When you write "fix", are thinking in terms of introducing or inventing?

From the initial fundamental rules, it is possible to derive (discover) new rules.

This is what happened after Efremovic introduced his rules for a proximity (nearness)

relation. Later, Efremovic's rules were refined by Smirnov.

There is still the issue about what fundamental rules represent? What is the relation between the fundamental rules of a system or a game and nature? Does nature enter into the formulation of a set rules?

Dear James

Then Smirnov discovered an emergent property in the virtual world invented by Efremovic. -)

1) Introducing or inventing ? I think both. Some of the FR can be newly invented for the current game; whereas others can be just borrowed from previously invented games.

2) Does nature enter into the formulation of the FR? That's a very good question and I see where you are getting at, because the answer may lead to the complete breakdown of my own point of view. Obviously, at the very fundamental level, the role of Nature cannot be excluded as we are, ourselves, part of Nature. It is very likely that anything we invent can be linked back somehow to something inspired (hence discovered) from Nature. For example, basic shapes are discovered in Nature then by asking ourselves "what will happen if ..?" we invent new shapes that we have never seen before. It is mostly our powerful capacity of visual abstraction that leads to invention.

@H.E. Lehtihet: Then Smirnov discovered an emergent property in the virtual world invented by Efremovic. -).

You have an interested idea. The axiom system for a proximity relation can be thought of a set of conditions for a relation a set of objects in a virtual world. In fact, if we modify the EF axioms in a descriptive approach to the nearness of sets of non-abstract points, then your idea of an emergent property would be found in the nearness of sets in the physical world. For example, let X be a set of picture points (a picture point that is a location with measurable content) and let X be endowed with a descriptive proxilmity relation. Then the Efremovic axioms can be transformed into a set of conditions with significant application in familiar settings such as palaeontology or even video games.

@H.E. Lehtihet: "Invention" comes from a latin verb meaning "to come upon" or " to find". "Discovery" refers literally to "uncovering something". Reading your definition of discovery only confirms that the two concepts are pretty close. In fact, in some dictionaries I see that "discovery" is used to explain "invention".

As to the question at hand, we should therefore make clear what is the *intended* meaning of each word.

The opinion, that a mathematical theorem is invented, suggests (crudely) that one just just came upon it like finding a silver coin on the street, the effort being largely to pick it up.

The opinion, that a mathematical theorem is discovered, suggests an amount of careful investigation that preceded and even caused the event.

If one respects the hard work which proving a theorem mostly involves, one cannot but prefer the second opinion, regardless of ones inclination (Platonism, Formalism, Realism,...)

Dear Marcel,

Thank you for your etymological precision. It is true that these two words are sometimes used interchangeably. However, my understanding is that the accepted meaning is that the object of an invention did not exist before the invention; whereas the object of a discovery did exist before the discovery.

On the other hand, when applied to mathematics (or to anything else for that matter), these words do not carry with themselves (not even crudely) any additional information regarding the amount of work that was required to achieve the invention/discovery.

In other words, when I say that A is invented and B is discovered, I do not mean at all that the invention of A was easy and that the discovery of B has required a hard work. I am not making any judgment regarding the value of a discovery or an invention. In fact, Most often, a discovery/invention is achieved only after a tremendous amount of work; albeit, sometimes, it can be achieved serendipitously.

I hope this has clarified my point of view. Thanks again.

Dear Marcel,

After reading your post, I started thinking about some possible pejorative meanings associated to "invented" and "discovered".

1) Suppose that someone invents a new product. Then, when reading about it for the first time in a newspaper, people could say they have "discovered and invention".

2) Now suppose I falsely claim that I have discovered a new planet in the solar system. Then people could say that I have "invented a discovery".

Therefore, we may "discover an invention" and "invent a discovery". In both cases, it would not cost us much in hard work; but it is true that in the second case there is clearly a negative connotation.

This is the only negative connotation I can think of concerning an invention.

It seems that I neglected the point of "(non-)existence before the act". But what if I compare the act of invention more explicitly with recognizing that the obscure grey thing at your feet is actually a valuable silver coin? Before this *creative act*, there is no thought of a silver coin in ones mind, but there is afterwards.

Also, I expressed rather crudely that you "only have to pick it up", suggesting unintendedly that invention is easy and goes without efforts. With the corrected formulation, the difficulty of invention lies in recognizing the value of something that most people just pass by without noticing. Allow me to explain this further.

If I remember correctly, someone named Stephenson *invented* the steam engine. Before that, anyone who boiled water should have noticed that the steam could lift up the cover. Stephenson's invention is *basically* to realize that this innocent-looking, somewhat disturbing, phenomenon could be turned into a useful force. And yes, the final act was to create a material object called a steam engine (which took most --perhaps nearly all-- of the efforts).

I think that the basic act of invention usually takes only an instant of deep inspiration; it is the putting to practical use that consumes most efforts.

Intuitively, I use the term "invention" to describe useful concepts in mathematics: E.g.,

the notion of "topological space" was invented to provide a solid foundation for dealing with limits and continuity. Or: the notion of differentiable manifold was invented to replace and generalize n-dimensional surfaces, which require a containing Euclidean space. The "putting to use" then corresponds with developing an extensive theory of topological spaces or of differentiable manifolds, making the promise being held.

In contrast, I prefer to use the term "discovery" for the act of formulating and proving a new theorem. Finding the formulation of the result-to-be-proved may sometimes take only an instant, but to make it a theorem may take an enervating sequence of efforts and desperation. At the end of the process, proof has been given and the truth of the statement is unveiled, according to the literal meaning of the word dis-covered, "Putting a theorem to use" is a different act, which may occur in addition. It means: to apply the result in some other discipline or in the real world. (The term "invention" could be justified here, too).

Dear Halim Lehtihet,

The bottom line is perhaps that the dictionary meanings of "invention" and "discovery" are confusingly close and, moreover, the material-world examples suggest that "invention" requires a coming into existence while "discovery" does not. That's a delicate point in the esoteric mental world of mathematics.

Your viewpoint has become clearer to me, and I hope I clarified mine after correcting a few careless statements. This is not a true-or-false discussion, but an exchange of opinions.

Dear Marcel,

I very much agree with you when you said that this is not a true-or-false discussion. We are just expressing our viewpoints. In doing so, we sometimes choose improper wordings that do not reflect exactly our thoughts, especially if English is only our third of fourth language. It is then the feedback of other contributors that help us clarify our views.

For example, after getting a few questions from James (see previous posts), I "discovered" that my initial view was not totally free of contradiction. Hence, thanks to James, I have learned something new, which helped me refine my own opinion.

Thank you Marcel

Not sure how to respond directly to answers within this thread @James Peters, but here goes ...

My initial reaction is that words are also collections of symbols QED (!) But that seems overly pedantic, so let's pretend I never said it. I think what I'm really getting at is that words are a pretty poor form of notation.

In fact I would go further than that, and suggest that whilst "assertions containing only words" may be provable theorems, they are not mathematical theorems.

The only one of the three I'm familiar with is Euclid - "There are an infinite number of prime numbers" - which is also the oldest and therefore perhaps the most instructive in terms of historical use of notation, so I'll focus on that if that's OK. It doesn't take long to 'discover' that he never actually proved the assertion in the familiar form you have stated it e.g. http://primes.utm.edu/notes/proofs/infinite/euclids.html

In fact he never proved the assertion in any form, mathematical or otherwise - because he never deals with the general case. Not only that, but the Greeks do not appear to have 'discovered' infinity and/or 'invented' a symbol for infinity (I would argue the latter prefigures the former), so would have been unable to write down a theorem with the word "infinite" in it.

I would contend that Euclid failed to 'invent' the necessary and/or sufficiently good mathematical notation to do so. For example there are no symbols representing the general case, no symbolic representation of infinity, and it even appears that the Father of Geometry was working without the benefit of integers i.e. symbols representing 1, 2, 3 etc., only the concept of a unit and measures corresponding to the numbers 2, 3, 4 etc. obtained by adding units together.

In other words Euclid cannot have 'discovered' a mathematical theorem about an infinite number of primes because he never 'invented' sufficiently good notation to do so!

@Halim: In general, "inventing a story" has a negative connotation of setting up something, suggesting fraud and swindle. I share your feeling that, besides this, "invention" seems to have positive connotations only.

@James: I considered your question "is a theorem invented or discovered?"

strictly within mathematics, excluding theorem-like statements about reality and disregarding the relationship of mathematics with the outside world.

By using verbs like "to invent" or "to discover", you obviously refer to the human activity of doing mathematics. The formalist attitude that

"A theorem in mathematics is really nothing more than

a collection of symbols that someone has taken the trouble

to write down in a meaningful fashion"

(where the word "provable" should be added) does not account for the greater part of the human efforts needed to produce a theorem. Therefore, one cannot expect it to provide reasonable answers to the question.

@James and @Halim: When a concept is formulated (say: a proximity space, or a topological space, a group, whatever) it can (at least in principle) be formalized in full detail to allow proofs withstanding all criticism. This implicitly fixes the "rules", as you express it.

To me, the rules are fixed in advance as the accepted rules of logic, but I agree with your intention. In this way, the design of a mathematical concept implicitly includes all the properties that follow directly or indirectly. This observation amounts to determinism and probably applies as well to the creation ("invention") of our universe at the time of the Big Bang. Then, much as with physics, human activity in mathematics amounts to progressively "discover" valid properties (by means of valid proofs, sometimes preparing the way with experiments, computations, or a search for examples).

I think this is a well-balanced view on pure mathematics as it is "performed" by humans. It seems justified, then, to think of concepts being invented and theorems being discovered.

In an extended view, including the relationship of mathematics with reality,

theorems need not only to be true (according to the accepted rules of logic) but they should not disagree with reality and preferably apply to reality. Concepts and theories that have no counterpart in reality may be considered as "supporting tissue" in mathematics.

An interesting concrete issue is about the "system of reals" appearing as a "continuum of numbers", which is required for traditional calculus. Many physicists believe that the universe is essentially discrete, which is the opposite of a continuum. This makes an intensive use of calculus in physics a bit suspicious (to say the least).

Another issue relates with the "observation" that a potential theorem may appear first in the mind of a researcher as an intuitive feeling about a subject, and a valid proof may obtain by adding details to a rough analysis of this intuition. This phenomenon suggests a rather intimate relationship of mathematics with physical reality. It may provide a different view on the nature of mathematical theorems.

@Richard:

I consulted the url you gave. The proof closest to Euclid's avoids both an argument by contraposition and an argument by contradiction.

Now I just wondered... I cannot recall any proof in Euclidean geometry using such arguments. Is it true that this type of reasoning was unknown (or rejected) in Euckid's time?

@Marcel Van de Vel: ...the notion of "topological space" was invented to provide a solid foundation for dealing with limits and continuity.

You may want to add that observations about the nearness of points and sets are formalized in topological spaces.

@Richard Hibbs: Euclid cannot have 'discovered' a mathematical theorem about an infinite number of primes because he never 'invented' sufficiently good notation to do so!

I agree with you, except for the last part. First, although Euclid did not have the concept of infiinity, Euclid is credited with the insight about prime numbers that led to the contemporary formulation of Euclid's result. Notice that we have a notation for cardinalities of infinity (thanks to Cantor) but we do not have a notation for a finite set, except by denial.

That is, a set A is finite, provided the cardinality of A is less than the cardinality of the natural numbers. In other words, it is possible to have a theorem about finite sets without having a notation for finitude.

Oh these sophisticated mathematicians of today! Are they aware that they need unprovable axioms to get hold of the set of natural numbers? Not to speak about cardinality!

Euclid would fall about laughing could he see what we made out of his clear and minimal argument.

I believe mathematical theories (that is, axiomatic bases and inference rules) are more or less "invented" to model something (quantity, space, structure, change, processes or reasoning). Theorems themselves, though, (provable statements about a theory) are discoveries done upon the invented theories, and proofs of a theorem could be considered discoveries as well.

Hard to know for sure, and I'm quite intrigued that this topic is much more heated than I thought it would be. Whether mathematical objects are entities of the universe or part of human reasoning is a debate that has been going on for years now, and it all narrows down to a matter of ideology in the end.

Conjecture: axioms reflect our perception of patterns discerned from signals for the senses. In other words, one can argue that axioms reflects our perceptions.

For example, here is an axiom for a proximity relation introduced by Efremovic during

the first part of the 1930s:

If a nonempty set A is near a nonempty set B, then B is near A.

The suggestion here is that it seems reasonable to write down the EF axiom after one has observed near sets in perceived patterns in nature.

The question here is this. Are patterns perceived by in the flurry of signals from the senses more fundamental than any axiom (or theorem) we might write down?

That's quite an interesting way of seeing things, I have to say. Axioms would be a formalization of what we can percieve, and theorems would be what we can infer from what we percieve, and thus qualifying as discoveries in that sense as well, or did I miss something?

@James

google did not allow me to find Efremovic's axioms. Would you mind writing them down for us? The symmetry property you mentioned can't be the whole story!?

@Ulrich

I found a pdf that contains Efremovic's proximity axioms in the first Google result I looked up.

They can be found here: http://www.kmo.or.kr/include/journal/downloadPdf.asp?articleuid=%7B2CA67C2A-126F-4B5A-8A89-F41B333149C2%7D

@Marcel

"Euclid often uses proofs by contradiction, but he does not use them to conclude the existence of geometric objects. That is, he does not use them in constructions. But he does use them to show what has been constructed is correct. "

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI6.html

I don't know enough logic to say if this also answers your contraposition point, but I do think the suggestion that proofs by contradiction (aka reduction ad absurdum - I'm sure you remember this!) do not imply "existence of .. objects" is interesting in the context of what we mean by 'discovery' in this discussion.

@Ulrich Mutze:

The best introduction to Efremovic's (EF) axioms is given by Smirnov (in Russian) in 1952, the same year that Efremovic's article (in Russian) was published. As far as I know, Efremovic's article has not been translated into English. Fortunately, Smirnov's article was translated into English by the Amer. Math. Soc. in 1964. The advantage in starting with Smirnov's introduction to the EF axioms is that they are given without symbols, which is not the case with the article found by Richard Hibbs.

My computer locked up this morning. Hopefully the problem with my PC will be resolved soon and then I can share with you Smirnov's rendition of the EF axioms.

@Ulrich Mutze:

Here are Efremovic's axioms:

EF.1 If the set A is close to B, then B is close to A.

EF.2 A union B is close to C, if and only if, at least one of the sets A or B is close to C.

EF.3 Two points are close, if and only if, they are the same point.

EF.4 All sets are far from the empty set.

EF.5 For any two sets A and B (in X) which are far from each other, there exists C and D (in X), X = C union D, such that A is far from C and B is far from D.

Hint: for EF.5, let D = complement of C.

Daniel Page

This would depend on ones conception and notion of "discovery". It would lean heavily on the question that others have answered on RG of "Is Mathematics discovery or invention?". I believe that new ideas are discoveries, but assertions are moreso inventions to find new theorems that may be discoveries. Let me give an example:

Suppose if someone figured out that there existed a polynomial-time algorithm (just knowing it exists, but may or may not have the actual thing) for a case of an NP-hard problem (not the whole problem, but maybe a subset of the instances), I would call this a discovery. Why? It requires a bit of intuition to figure out as nobody has figured that out yet. Not to mention that it is proven. I'd not call something a theorem unless it was.

Now, what I'd call an invention is an axiom that was boldly asserted to permit new discoveries. For example, I'd not call Peano's Axioms a discovery but an invention. Whether a computer (as in computation) can figure something out or not should not lean on the notion of a new idea. If somebody has not figured something out, it should be called a discovery as it is a contribution to what we know now. That being said, there are varying "degrees" of value in discoveries. Some are trivial while others require a lot of intuition to figure out. As a consequence of this, something new to understand (not assert) should be in the realm of knowledge, not engineering and design as an invention. We apply what we know to construct inventions (which is quite a bit different than the concept of proving something then yielding a result formally).

In short: I find it valuable to distinguish what we make up from what we reason to be true. I wouldn't call something that is a new contribution to the set of knowledge an invention if it were asserted. Theorems are proven to be true (whether it be through axiomatic methods, construction methods, or intuitive methods.).

Hope this helps!

@Fabricio and @James:

Axioms are a set of statements providing what we think is essential information on some conceived "world". The term "model" coins this view on "axiom system" pretty well, although formal logic reserves this term for the specific structures (the examples) that obey the properties described by the axioms.

In restricted situations, a choice of axioms may be driven by sensory perception. Quite often, however, axioms are a result of retrospecting certain proofs, with the intention of making an inventory of their vital ingredients. (Prominent examples are: metric space, topological space, group, field, ordered set...) A great deal of math has a rather weak relationship with sensory information. When it does have such a connection (as is the case with, e.g., proximity spaces), one should realize that examples may stretch far beyond the world of our senses.

All contributors seem to consider as selfevident that theorems result from axioms. There is a second option: they may result from constructions (constructive mathematics). Of course, also constructions need some agreement concerning allowed means and methods of proof. But these clarifications don't fall under the category of 'axioms'.

@Ulrich: Maybe inference rules? No, nevermind, definitions are not always built upon inference rules, and many theorems are built upon definitions (that is, if a certain object has a set A of properties, and some of these properties imply a set A' of properties that make it fit the definition of B, then it is a B).

@Marcel: You kind of have a point, but when James said "perception", I don't think he was limiting himself to sensory stimuli, but speaking about our intuitive understanding of the behavior of a variable as a whole.

@Fabricio Kolberg: ...our intuitive understanding of the behavior of a variable as a whole.

Agreed. The question of discovering theorems (and axioms) relative to perceived patterns in nature leads us back to the original issue. From perceived patterns, we can conjecture what perceived patterns are possible. Recall the saying from the Greeks: from possibility, infer existence. Theorems that reflect conclusions derived from the perception of the possible, we user in theorems that reflection invention.

The last line should read:

Observations that reflect conclusions derived from the perception of the possible,

we use in theorems that reflect invention.

Going back to the EF axioms, notice

EF.1 If the set A is close to B, then B is close to A.

This axiom implies a spatial relation between A and B, i.e., sets A and B are spatially close. Closeness needs to be defined. Traditionally, closeness means either the boundaries have a common point or that the intersection of the two sets is not

empty.

The problem here is the relation between a set of points in mathematics (such points are abstract, dimensionless and just have a location), whereas the points in a perceived pattern are non-abstract (called a place with a place and with measurable content).

This leaves open the question of how we define closeness of non-abstact points in perceived patterns derived from sensed signals.

Nitish Ranjan

According to me theorems in mathematics are discovered and invented both.

There is a slight difference between Discovery and Invention.

Discovery is finding out something that preexists, while invention is using objects that preexist to create something new that is first of its kind.

In Mathematics, pattern preexists so finding that pattern is discovery and in theories they are simplified into new form for easy understanding so it can be said as an invention.

Regards:

Nitish

Dear James,

The following is slightly off topic and might deserve a thread on its own.

When you speak about a set of points and perceived patterns, it reminds me of the work done to study human performance in solving the symmetric case of the Euclidean TSP. See for example this paper by James MacGregor as well as some other papers that are cited in there:

http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1140&context=jps

In turn, algorithms have been proposed based on how humans solve the problem. See for example :

http://arxiv.org/pdf/1303.4969.pdf

@Nitish Ranjan:..t.heorems in mathematics are discovered and invented both.

I agree.

Here is a thought problem, leading to a theorem.

Small things that we see with a microscope usually have small diameters. For example, a microfossil sometimes found in amethyst can have a diameter of roughly 5 mm. In a diameter measuring 5 mm, there are a finite number of picture points (pixels) in a microscope image. What is the difference between a line segment in a microscope image vs. a line segment in the Euclidean plane? Consider

Theorem Let X be a set of points in the Euclidean plane and let L be a line segment in X mearsuring 5 mm. There are an infinite number of points in L.

James,

probably unintentionally you brought to mind an interesting fact: you can't define 5 mm within mathematics!

@Ulrich:

When I prove a result in Discrete Mathematics, axioms often seem far away. But behind the curtains are lurking the laws of natural numbers with the principle of natural induction, and also some principles of set theory.

Use of "axioms" (through inference rules) may suggest a high degree of formalism,

but it is essentially common-sense reasoning which can be formalized to any degree if wanted. A construction obeys the same rules as it involves verification to make sure that it behaves as intended.

Before the 20th century, reasoning in mathematics was entirely based on common sense and real-world experience. This occasionally lead to disputes about "stretched common sense", as with infinitesimal calculus (formalized nowadays as "non-standard analysis"). It also lead to unnecessary disputes about whether "zero" is a number or about the "existence" of imaginary numbers.

Essentially, reasoning nowadays is not much different from what it was before and persistent disputes have lead to the acceptance of certain "deviating views" like intuitionism and constuctivism.

Different viewpoints in this thread seem to depend on how much "real world" you include in your mathematical *practice* and how you evaluate the formalism that founds your mathematical reasoning.

Marcel,

I agree with all you write. So is the world of mathematics, but, so my thinking, it should not be so (I know Russel (or Shaw?) 'a reasonable man does not want to change the world').

Books on classical mathematical logic (e.g. Ebbinghaus,Flum,Thomas) drive me gracy by numbering variables v_1, v_2, ... and proving later that first order logic can not define the natural numbers. In my eyes, common sense would ask for a definition of natural numbers (e.g. as words over a one-character alphabet) first, and then coming to terms, expressions, and all that stuff. When comming to first order logic, when writing down axioms (e.g. for groups) common sense would ask for an explanation of the 'for all x,y,z ....' (e.g. (x.y).z = x.(y.z) ) saying that we don't yet know what these x,y,z will be and that the whole appoach to get hold of mathematical structures by this method is one way, and the direct construction (such as that for the natural numbers that I refered to earlier) is an other way. It is amusing to see that all methods needed for instance to construct the universe of the hereditarily finite sets (and the necessary inductive methods of proof) are made available for handling terms and expressions but some kind of professional blindness hinders most mathematicians to recognize that these can be used to define the most basic mathematical objects such as numbers directly.

Unfortunately, everybody who knows the scene can tell you dozends of approaches that go just along these lines, but nobody would see the need to adjust the mental building that mathematics provides as an interface to the applied sciences (such as physics) in a way that does not offend common sense so harshly as it presently does.

Marcel Van de Vel: Use of "axioms" (through inference rules) may suggest a high degree of formalism, but it is essentially common-sense reasoning which can be formalized to any degree if wanted.

Yes, I agree. Good examples of axioms that reflect common-sense reasoning the axioms for a proximity relation (Efremovic, first part of the 1930s) and for a tolerance relation (introduced without axioms by Poincare in 1895 and with axioms by Zeeman in the 1960s).

@Ulrich: The "non-logical axioms" of a deductive system are supposed to account for things that cannot be defined by its logical axioms and inference rules alone (or for things you don't want to bother proving from the basic axioms), and that's what is normally used to define the set of natural numbers inside a formal deductive system.

@Ulrich: (in defense of formal logic)

Not only do they number variables as v_1, v_2, ..., they also define things like well-formed formulas, interpretation in a model, and truth in a model, by mathematical induction on the complexity of expressions.

Numbering variables is a harmless habit providing an infinity of symbols without using any number theory. (Most authors switch afterwards to the usual x,y,z because that's what you see in practice.) The point is, that all this is "meta-mathematics", which is common mathematics on a different "communication line" with the reader. The proof of the pudding is, that all these definitions can be put to work (in principle) by a smart computer program. An array of pointers effectively numbers the variables, and the inductive descriptions can be translated into recursive procedures that recognize the language or build up formulas.

Keep in mind that the whole of mathematics is basically a language to communicate exact statements. Although we like to think of mathematical objects as things of the mind, or even better, as real objects, they are communicated rather than handed over. This might not be your (or my) daily attitude towards math, but in formal logic you must be prepared for this confronting fact.

One other thing: formal logic can shed a different light on certain problems or deal with problems you won't come upon in other circumstances.

One famous example is non-standard analysis, rediscovering the good old "infinitesimals" used up to the 19th century. Infinitesimals do occur if you cleverly switch between two models of the real line and effective reasoning is possible with them.

For a more modest example, see my 2009 paper "Theories with the Independence Property" (available on RG), dwelling on an "independence property" observed in the eighties/nineties by Artificial Intelligence people in the area of automated problem solving and language design. I considered the phenomenon in general first order logic and found that the property goes with a certain type of axioms. This lead at once to a massive number of "theories" possessing the property: group theory, ring theory, Boolean algebra, Tarski's relation algebra, and many others, even Pasch-Peano geometries. Before this (to my knowledge) mathematicians never observed the phenomenon in any of these theories, although some of its consequences (e.g., free models) were known in some cases.

I assume this is getting away from the subject of this thread... (it's a rich subject).

Marcel Van de Vel: (in defense of formal logic)...

Actually, formal logic and its usage in proving theorems brings us to the threshold of the invention side of mathematics.

Ulrich Mutze: ...all methods needed for instance to construct the universe of the hereditarily finite sets (and the necessary inductive methods of proof) are made available for handling terms and expressions...

I definitely agree.

Penultimate examples of the use of finite sets in working toward discoveries in mathematics come from Leibniz (reasoning about integration), Cauchy (reasoning about continuity), Hadamard (geometry) and the work of Frechet (Hadamard's student), leading to metric spaces in Frechet's doctoral thesis, 1906.

@Ulrich:

(quote) Oh these sophisticated mathematicians of today! Are they aware that they need unprovable axioms to get hold of the set of natural numbers? (unquote)

Euclid attempted to "define" the undefinable (because primitive) concepts of point (without size) and line (can be continued indefinitely and has no thickness). His major aim was probably to invite his audience at a minor idealization of these real objects. Similarly, his "numbers" were considered a result of measurement against a fixed unit length.

Modern formalisms (e.g., set theory) do not attempt at defining the primary concepts (set, membership), but fix the "rules" by which they behave through an "unprovable" axiom system on undefined (meaningless) objects.

This vital abstraction opened the way to progress in mathematics.

The discovery of non-Euclidean geometries (solving Euclid's problem of the parallel postulate) is mainly due to the insight of having different models of one axiom system,

not being bothered by a "realistic" idea of "straight line".

The foundation of modern calculus lies in a well-balanced concept of real numbers forming a complete and totally ordered field, not in measurements with a fixed unit length,

If Euclid would be back among us today, he might exclaim "why didn't I think of this?".

He would certainly not laugh with what we made out of his "clear and minimal arguments".

Marcel,

there is no need to defend the body of formal logic, at least not against me. My critic concerned exactly the points I brought up, I need not to repeat them. Instead, I may explain more fully how I consider the fundamentals of mathematics could (and in my mind should) be presented: define (hereditarily) finite sets with the formal means that allow us to speak of terms in a formal language (that would include the definition of natural numbers). Then you have a basis that supports everything that on a computer ever can be done. On the other hand, it is such basic stuff that a practical person may forget it, as most people forget the fundamentals of arithmetics (if ever they knew them). Then the more mathematical topics dealing with infinities and limits may be added to solve problems for which they are needed. (It's not easy to find such problems). The main part of mathematics would be built on the finite set ground (here is the element relation in sets is computable since each set is a finite series of symbols that provide input for the decision algorithm). Many existing approaches to constructive mathematics could guide the way for producing a demystified version of most of mathematics.

My remark concerning the laughter of Euclid, concerned only the arguments in the present thread that a concept and a symbol of infinity would add anything significant to Euclids insight that no (of course finite) list can contain all primes.

"...to characterize the import of pure geometry, we might use the standard form of a movie-disclaimer: No portrayal of the characteristics of geometrical figures or of the spatial properties of relationships of actual bodies is intended, and any similarities between the primitive concepts and their customary geometrical connotations are purely coincidental." (C.G. Hempel)

I wonder if the above quote has any relevance in the context of the original question of the present thread.

H.E. Lehtihet ...any relevance in the context of the original question of the present thread.

Strangely enough, yes, if we consider the arcs and lines in pure geometry as analogues of the arcs and lines an observed scene in nature. That is, it can argued that pure geometry is an invention inspired by observed patterns in sensed signals. But it seems that C.G. Hempel would deny this.

This is of particular interest because the geometry of 19th century led to the discovery of topology by Hausdroff and proximity spaces (called continuous geometry by Efremovic) in the early part of the 20th century.

Thank you James. I was asking about the relevance of this quote because I do not know its exact context.

Food for thought:

If you’re looking for a side to join, then maybe the Platonic theory is your cup of tea. The Classical Greek philosopher Plato was of the view that math was discoverable, and that it is what underlies the very structure of our universe. He believed that by following the intransient inbuilt logic of math, a person would discover the truths independent of human observation and free of the transient nature of physical reality.

“The abstract realm in which a mathematician works is by dint of prolonged intimacy more concrete to him than the chair he happens to sit on,” says Ulf Persson of Chalmers University of Technology in Sweden, a self-described Platonist.

And while Barry Mazur, a mathematician at Harvard University, doesn’t count himself as a Platonist, he does note that the Platonic view of mathematical discovery fits well with the experience of doing mathematics. The sensation of working on a theorem, he says, can be like being “a hunter and gatherer of mathematical concepts.”

Mazur provides the opposing view as well, asking just where these mathematical hunting grounds are. For if math is out there waiting to be discovered, what once was a purely abstract notion then has to develop an existence unconceived of by humans. Subsequently, Mazur describes the Platonic view as “a full-fledged theistic position.”

Brian Davies, a mathematician at King's College London, writes in his article entitled “Let Platonism Die” that Platonism “has more in common with mystical religions than with modern science.” And modern science, he believes, provides evidence to show that the Platonic view is just plain wrong.

So the question remains; if a mathematical theory goes undiscovered, does it truly exist? Maybe this will be the next “does a tree falling in the forest make any sound if no one is there to hear it?”

http://www.dailygalaxy.com/my_weblog/2008/04/is-mathematics.html

Food for Thought, indeed.

The views on Platonism of Mazur (theistic) and Davies (mystical religion) seem to refer to the original formulation of Platonism: forms or ideas constituting the primary existence, a world of the gods, in contrast with "reality" as created by a low-ranking demiurg. I cannot believe that this thinking is still taken seriously by modern people. The Platonist view on mathematics --with myths of gods and their creations left out-- remains attractive to this day.

Several answers in this thread underpin the view that, given the rules of predicate logic, once a mathematical concept is formulated, the scene for discovery has been set. In this way, the status of mathematical theories is somewhat the same as the one of physical laws. What is the status of undiscovered laws in physics?

So it seems to me that we should rather focus on the status of mathematical concepts and their "(non-)existence before being invented" for a truly novel question.

One more thought.

There is some dissimilarity with the "falling tree" problem: In the absence of humans, a falling tree still goes with vibrations of the air, which would have been understood as "sound" otherwise. A (sensible) concept lies awaiting at least as the formal string of symbols that expresses the concept, but it takes more than a coincidental human presence to perceive it as a useful concept.

Marcel,

For the question: “does a tree falling in the forest make any sound if no one is there to hear it?”,

we can put this Bishop Berkeley puzzle to rest by putting a sensor near a tree to verify that makes sounds when it falls, even someone is not present to witness the sound of the falling tree.

@James: nice remark.

As to concepts in mathematics, it normally takes a skilled mathematician to read a useful concept from a formal string of logic symbols. Automated proof software could start drawing conclusions from a (randomly well-formed) expression and qualify it as useful by using quantitative criteria.

Actually, I'm not fond of the idea that concepts (and theorems) pre-exist as formal expressions. It is reasonable to think that the persistent focus of mathematicians makes mathematics into a kind of reality coexistent with physical reality, not the formal expressions which usually come afterwards.

"The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight." (D'Alembert)

Layal Hakim

I think it depends on the theorem.

For example, I would say that Pythagoras' Theorem (and most relations in Euclidean geometry) was discovered because you can't falsify the calculations and using measurements we can see that it holds.

However, one thing I always ponder about are theorems about complex numbers. I mean, the complex number ''i''... wasn't that invented? So, all the theorems which use the fact that i=sqrt(-1), do they crumble if this invention wasn't made?

Layal,

Consider not only sort(-1) but also the introduction of symbols that lead to the discovery of theorems. The classical example is the integral sign, which is generally attributed to Leibniz. It was Leibniz who wrote his mathematics long-hand (cursively) in Latin and used an elongated S to represent his idea of an integral. Just think about all of the mathematics (and theorems) that grew out of the introduction of the integral symbol.

The imaginary unit "i" is essentially a piece of information, a rule of behavior: it is an object that squares to -1. It occurs fairly naturally by interpreting the Cartesian plane. My favorite construction of "i" comes from algebra: take the ring of real (formal) polynomials in the variable X, R[X], modulo the polynomial X^2 + 1. Then "i" is the class of the symbol X. This is a special case of forming quotient rings (just like "the integers modulo a given number m" -- clock reading if m=12). The process of forming quotients comes essentially from set theory, and it is a jewel of mathematical thinking.

The imaginary unit is therefore not "a historic accident of thought", but the outcome of several roads of thought. If "i" hadn't been invented in late medieval times, it would most certainly have been later on.

What distinguishes mathematics from "fiction" (as in literature, sensory or mental illusion, wishful thinking) is its extreme degree of consistency and concurrency (not to mention an undeniable degree of realism).

Theorems need language means to express them. These language means in most cases are clearly invented. They may be invented in different versions (e.g. Weierstass's and Riemann's view of 'analytic functions'). The theorems themselves, in most cases are clearly discovered: If the complex arithmetic has been set up by invention (an invention guided by observations and suggestions from many sources as Marcel pointed out) we can not i n v e n t a theorem like exp(i*pi)+1=0. We have to discover it (or learn it from the great discoverer Leonhard Euler).

Happily invented language means in mathematics (e.g. integrals, derivatives, complex numbers, transcendental functions) create questions and the freedom to tackle those that seem to be within one's intelectual reach. The Bernoullis and Euler saw so many problems springing out of the basic idea's of calculus as put forward by Leibnitz and Newton that they hardly could get enough paper to work them out. David Hilbert saw himself in a 'paradise' created by Georg Cantor's set theory . In the past few decades the simple idea of a 'dynamical system' (inherent to physics from its beginning) under the influence of the computer became a field that allowed easy hopping from problem to problem and nevertheless promised deep discoveries.

Platonism seems to stress the point that what can be invented and what creates the freedom felt by any involved scientist, is actually not free but is to be chosen from a predetermined pool of 'essential mathematical concepts'. Since the validity or absurdity of such a concept has no real influence on what mathematicians actually do, they may hold oppinions either way.

Tolga Soyata

I would say, BOTH, but, with a painfully high percentage, most definitely "invented."

INVENTING is adding another brand to our knowledge tree based on what we know.

DISCOVERING is adding a completely new branch to the knowledge tree.

Since we build the knowledge tree on top of the existing one, we BIAS ourselves based on what we know and this is what restricts our improvement. So, we are highly dependent on how the knowledge tree was formed before us. It is extremely rare that, a scientist DISCOVERs a brand new concept ... May be the invention of a new element on the periodic table. This is a DISCOVERY, I believe.

Take Fermat's Last Theorem as an example : a^n+b^n=c^n is not possible for any integer a, b, c, n ... The proof was INVENTED by Andrew Wiles a couple decades ago from today. But, Andrew Wiles knew Elliptic curves (EC), Fermat didn't. So, that branch of the KNOWLEDGE TREE didn't exist, making it impossible for Fermat to even imagine a solution using EC's. Although Fermat claimed that, he had a solution, and this puzzled mathematicians for 400 years, we won't get into that :)

So, let me go even further back, when was even the mathematical symbolism introduced ? It is Diophantus ... Before Diophantus, all humans had was "things that are equal to each other are equal to one another" !!! NOT if a=b and b=c --> a=c !! By having strictly WORDS to introduce theorems, can you even imagine inventing anything sophisticated ?

My point is, we INVENT things based on what we already know, i.e., we try to FIT a new theorem on top of the knowledge tree we have ... Assume for a second that, Diophantus invented COMPLEX NUMBERs. Would it take all the way to 1700's for Euler to invent --- e^ix = cos(x) + isin(x) ? Or, would we invent it 1000 years before that ?

Is there any guarantee that, the knowledge tree we built so far is not a rotten, bug-infested one ? and, this is what is restricting us from discovering a lot more ?

Peshawa Jammal Muhammad Ali

@Tolga

you missed to add the essential conditions a>0,b>0,n>2 in your formulation of Fermat's Last Theorem. About Fermat and elliptic curves I read in a nice booklet 'Translations of MATHEMATICAN MONOGRAPHS Vol 186' Number Theory 1, Fermat's Dream by Kato, Kurokawa, and Saito, p. 19: "Studying integral and rational points on an elliptic curve was Fermat's favorite theme, ...". By the way, there is nothing difficult with the concept of an elliptic curve. The difficult part (which I did not master so far, although I did scan through this booklet) is to see how these innocent curves can be of any value for the problem under consideration.

I read this long conversation and it was interesting to me, but still my answer is math is discovered not invented.

For example the pi which is a constant number 22/7 is discovered because it was their on circles even before being discovered. Also all rules related to trigonometry (sines, cosines, tangents.....etc. and their related rules) are the best examples of discovery of mathematics....

@Peshawa,

do your few examples in favour of 'discovered' , according to your logic, exclude that for other examples one could vote for 'invented'?

Fairouz Bettayeb

At the beginning there was numbers, sets of objects (formalized by numbers), geometries (shapes from nature) and mental Logic (deductive, inductive, inference, formal/informal…etc). Consequently the first mathematical laws are discovered following mental logic correlated with objects and geometry in the environment. The mathematical knowledge goes on growth into a mental or experimental diversion process of assembling/ building /connecting of geometries or sets. At that time a language was needed to formalize these discoveries, which gives into birth the invention of symbols, predicates, assumptions, proofs, functions... Etc...

So mathematics is discovered following mental logic game or curiosity or life need, subsequent to a mathematical knowledge tree, and formalized by an invented language. As a result, which is invented in mathematics is the mathematical language.

Fairouz Bettayeb

Yes, i do think that mathematics is discovered and still continue to be discovered following mathematical knowledge tree, and its language was invented and will continue to be invented too..

@Fairouz Bettayeb :

Perhaps, in addition to the language of mathematics, you may want to add that the axioms, theorems and symbols of mathematics continue to be invented.

Michael Brückner

May I start from a potentially new point of view in this thread (not having read all posts, I humbly admit)? If you favour the idea of mathematics being invented, what hinders the vulture capitalists to suck it dry by claiming patents on the then precious 'inventions'? I do not like this scenario at all.

Would it not be similarly appropriate to talk about *developing* axioms, theorems, symbols and ideas of structures that quite often appear in nature and can thus be described as some kind of 'reality'?

As a physicist (from education) I have been grateful for the mutual stimulation of math and physics that have taken place in recent centuries. I view this as a continual process, a development of ideas.

I agree with Michael Brückner in that considering theorems as inventions may lead to corporate pigs attempting to patent them, making tons of money at the cost of a decrease in research quality all over the world (after all, I don't know many people who'd like to pay in order to cite someone in a publication, and I know of many people who would receive an avalanche of lawsuits were mathematical results to be patented).

Given that whether a theorem is discovered or invented is a matter more closely related to philosophy than mathematics itself (and therefore, doesn't have one single right answer), I'd say that it isn't dishonest to simply side with the most advantageous answer to the research and application of mathematics, at least legally speaking.

@Michael Brückner:

Good post!

Here is a related post from an attorney, who has consider the issue patenting mathematics:

The problem is not whether algorithms are discovered or invented, it is that they are often unique. Imagine if Fourier analysis was patented: literally all of modern digital a/v formats would be owned. What a hideous and absurd outcome. To put it another way, the difference between a patentable invention and an algorithm is configuration: should you be able to patent a particular hardware method of Fourier analysis? Maybe - I can see the argument. Should you be able to patent the math behind Fourier analysis? No.

http://www.reddit.com/r/math/comments/1am774/should_business_be_allowed_to_patent_mathematics/

Yes this is a helpful digression. If, as I argued much earlier in this thread, the 'invention' of good notation facilitates the 'discovery' of mathematics, then it is certainly possible to conceive of ways of certifying the novelty and ownership of the invention e.g. by recognising Intellectual Property Rights (IPR) entailed in the symbolic representations on the printed page, cave walls etc.

Patents are almost certainly not the appropriate method, from my limited experience of trying to commercialize intangibles. But instead as our Martial Arts colleague ably demonstrates (with considerable agility!) copyright is actually the de facto method in use today.

(And copyright, I might add, in response to some of the earlier contributions to our debate, is no laughing matter!)

Demetris Christopoulos

Dear James, your question is great, but I' d like to contribute to the debate by asking something more: Why does the Universe (Definition: The Set of all local universes like ours) allow us to describe it with so many equivalent and different mathematical ways?

@Richard Hibbs:

Here is a followup to your excellent post:

Intellectual property (IP) is a legal concept which refers to creations of the mind for which exclusive rights are recognized.[1] Under intellectual property law, owners are granted certain exclusive rights to a variety of intangible assets, such as musical, literary, and artistic works; discoveries and inventions; and words, phrases, symbols, and designs. Common types of intellectual property rights include copyright, trademarks, patents, industrial design rights, trade dress, and in some jurisdictions trade secrets.

Although many of the legal principles governing intellectual property rights have evolved over centuries, it was not until the 19th century that the term intellectual property began to be used, and not until the late 20th century that it became commonplace in the majority of the world.[2] The British Statute of Anne (1710) and the Statute of Monopolies (1624) are now seen as the origins of copyright and patent law respectively.

[1] Intellectual Property Licensing: Forms and Analysis, by Richard Raysman, Edward A. Pisacreta and Kenneth A. Adler. Law Journal Press, 1998–2008. ISBN 973-58852-086-9.

[2] Brad, Sherman; Lionel Bently (1999). The making of modern intellectual property law: the British experience, 1760–1911. Cambridge University Press. p. 207. ISBN 9780521563635.

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

@Demetris Christopoulos: ...Why does the Universe (Definition: The Set of all local universes like ours) allow us to describe it with so many equivalent and different mathematical ways?

Your definition is neat! That is, let the universe be a set of local universes like ours.

One answer to your question is this. The local universe we live in is susceptible to many different points of view, ranging from intuitive (imaginative) views (cartoons, poetry, fiction) to verifiable views (science: e.g., astronomy, astrophysics, chemistry, geololgy, palaeontology, physics, mathematics: e.g., set theory, topology, stochastic analysis, non-fiction: biography, history, travel). Each of these aspects of these points of view reflect our observations about the environment, natural phenomena, and social interactions. And each of these points of view has a relational model that describes what observe. The choice of relational model that describes what we observe appears to be endless. Nature has lots of depth and lots of facets that provided an abundance of mathematical perspectives.

Instead of speaking in terms of "equivalent", I suggest using the term "similar" mathematical ways. The similarity of mathematics models for what we observe in nature stems from the fact that natural phenomena tend to be descriptively near each other. In face, just as Martin Kovar has observed, it is possible to introduce a Topology of Everything that distils the essentials of what we observe. Added to this, it is possible to introduce a Proximity Space view of Everything, based on our observations about whether sets of natural phenomena are either near or remote from each other.

Fairouz Bettayeb

I agree with James "algorithms are often unique", for a same solution of a problem solving, many configuration of algorithms are possible; the best is one that gives more interesting outputs by a further processing. I think that what could be patented into algorithms (usually a set of theorems and info processing) is the logic behind the algorithm when it is quasi different from the other known solutions and brings new knowledge or finding into a specific field of science

Demetris Christopoulos

My point of view is this: If either local universes or their union (Universe) can be explained accurately with a great variety of mathematical ways, each one involving its axiomatic structure (definitions, theorems, lemmas, corollaries etc), then the question about discovery or invention of a theorem is transformed in the question of how countable (or not) are the similar ways that describe a certain phenomenon.

Juan-Esteban Palomar Tarancon

@Demetris Christopoulos:

Good post! From what you have written, an index to the richness of our perception of a local universe is represented by

1-to-1 correspondence between the number ways to describe a certain phenomenon and the members of a subset of the natural numbers.

Let f be a mapping from D into the natural numbers, where D is the set of descriptions of a certain phenomenon. For example, let d in D be a description. For the transformation of a theorem, construct the mapping

f(d) = i, where d is a description in D and i is a natural number.

Then obtain |f(D)|, the cardinality of the number of ways to describe a phenomenon.

Demetris Christopoulos

So, a good task for Science is to find the above cardinalities defined by James and not to fight about which description is better than others. But this task has a requirement: To overcome our scientific egotism.

The topic of this thread is more complex than it seems at the first glance. If you watch a scene in any street, for instance, a dog bites somebody. You can write this scene, and you have not invented it. Now suppose, that some scene you have never seen, appears in your mind in a dream. Suppose that you write the scene. Can you say that the scene is your invention?

For instance, the most popular piece by the great Italian composer Tartini, is "Devil's Trill Sonata." He said that Devil appeared in one of his dreams, took his violin and played it. When Tartini awoke, he remembered the melody and wrote it. If Tartini's explanation is true, was he the inventor or the discoverer of this piece? Of course, it was a product of his mind. However, what is the machinery? Are dreams caused by will or by external information stored in mind?

Perhaps, there is a fuzzy region separating inventions from discoveries. When we are handling examples in this fuzzy region, it is a personal preference to consider "invented" or "discovered". Personal preferences do not arise from logic; hence there is no logical explanation for them.

Juan-Esteban Palomar Tarancon

Dear Juan-Esteban: ...Perhaps, there is a fuzzy region separating inventions from discoveries.

Yea, good observation. It does seem that our findings are often partly invented and partly discovered. I am thinking of the classic characterisation of a discovery by a scientist who admitted that his findings were gotten thanks to his standing on the shoulders of giants. Was it Isaac Newton who made that observation?

Layal Hakim

Yes. Isaac Newton wrote a letter to Robert Hooke with a sentence included saying:

"If I have seen farther, it is by standing on the shoulders of giants."

Dear James,

It seems a case of telepathy. I was thinking something like. For instance, when a researcher "invents" some mathematical construction, in general, he applies some patterns learned from the work of the preceding generations. Each new idea and construction is partly obtained from the ideas of others, partly from the proper creativity and partly from the real world problems. Each part is linked to others through a fuzzy glue. A good example is a paper of yours. I can see very original constructions in it together with references and some pictures from the real world.

In my previous message, my main interest consisted of introducing the fuzzy logic in this thread. I think that it is a good practise because frequently we go from false to truth ignoring the way between both extremes.

Here is a story related to Newton admission about standing on the shoulders of giants.

This story comes from Greek mythology. A blind giant named Orion carried his servant Cedalion on his shoulders, so that Cedalion could see more of the world.

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

Layal and Judith,

Many thanks for digging out the story about Newton. I just now found the reference:

Newton, letter to Robert Hooke, 5 February 1676, The Correspondence to Isaac Newton, volume 1, Edited by H.W. Turnbull, 1959, p. 416.