It has long ago been established that math is, indeed, one of the few ways by which we human beings learn to think. Most sciences and disciplines just help us learn to know. By and large this is one of the most salient feature of (the history of) mathematics. The very growth of mathe concepts is one conspicuous way by which we can appreciate how reality has, correspondingly, been deeper and wider.

One more salient feature of math is that it allows us to think abstractly. Only a handful sciences do that, too. No individual, no society, no culture is truly... human and civilized if he or she does not think abstractly.

Dear James, thank you so much for your question. I love it!

It has long ago been established that math is, indeed, one of the few ways by which we human beings learn to think. Most sciences and disciplines just help us learn to know. By and large this is one of the most salient feature of (the history of) mathematics. The very growth of mathe concepts is one conspicuous way by which we can appreciate how reality has, correspondingly, been deeper and wider.

One more salient feature of math is that it allows us to think abstractly. Only a handful sciences do that, too. No individual, no society, no culture is truly... human and civilized if he or she does not think abstractly.

Dear James, thank you so much for your question. I love it!

There is an interesting - already forgotten? - paper of Eugen Wigner about the unreasonable effectiveness of mathematics. See also link to wikipedia.

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

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Sooner or later you will find some opposing views so I better add them already here ;-))

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

http://eprints.biblio.unitn.it/685/

http://www.amazon.com/dp/1447111214

@ Carlos. Nicely said. Math is a way by which we learn to think, whereas most sciences and desciplines just help us learn to know. 

Although by and large I would agree, I nevertheless wonder, what the crucial distinction is between thinking and knowing as meant here. Could it be that the words "thinking" and "knowing" as used here are just other labels for deductive versus inductive reasoning, or knowledge creation? If that is the case, then I would argue that inductive reasoning leading to new potential facts and insights, call it knowledge if you like, is even more fundamental than deductive reasoning, as it adds to our stock of knowledge (also about how we do deductive reasoning, i.e. thinking), whereas deductive reasoning is just verifying that knowledge and finding implicit facts in that stock of knowledge, sometimes pointing at missing pieces which can be found again by inductive reasoning. Or am I missing the point of James' original question?

A question of this extent can get only partial answers. What I propose now is even less then a partial answer. It is just an idea of this very moment, that I propose for discussion. I think that the origin of mathematical concepts can be found in toys, games, puzzles. Analogic arguments: children playing, one says "suppose that this is the King..." - mathematicians playing, one says "let's consider a triangle". Biologic argument: in nature, ontogeny repeats philogeny (the development of an Individuum. as an embrio, repeats the development of ist species...) Why not in spirit, the development of Individuum, who starts by playing with toys, should not represent the development of sciences, which are the experience of ist species. Finally, a semantic argument: toys are symbols for reality. You have there the car, the postman, the princess, Barbie, etc. It is the first step in abstractization. OK, now I am waiting for other opinions. Bye!

I completely agree with you dear Mihal. Playing is, for sure, at the very seeds of maths.

Many stories vcan be tod in this sense as proofs and arguments. One very recently I love to mind is, as you may know, D. Coxeter's playing as a source for all his discoveries about Coxeters sets, polytopes and many others - as Coxeter himself once told.

At the ontogenetic scale I do think that playing serves as a sort of craddle for good maths - over against the very much story about accounting, trade, etc.

Dear James,

You brought a good thread for discusssion. Mathematics is an ctivity performed through abstract thinking in search of perfect solutions to problems, as abstract thinking and precision are innate to mathematics. Since time immemeorial, mathematics  was made to solve problems encountered by humans that were related to numbers, shapes and geometry, continutity  and change, things that were relevant to their livevlyhoods.

We saw forinstance how the concept of zero was uknown for so long because they did not see the necessity of it,  the negative numbers were looked in suspicitions by mathematicians for long, but both were found essential to be created  to address problems of algebraic types. From your prelude, the development of curvature as a tool was to describe and study behaviors of curves and compare, which one is able to twist faster than the other or how a curve changes geometrically along its path, we saw how topolgy was initiated to solve and address real problems of connecttivity. 

The imaginary unit  i = √(-1) from the theory of complex numbers  was dislked and considered unreal  for so long, a concept appeared because of unsolvable algebriac equations in the set of real numbers. It appeared in algebraic solutions for equations written for wrongly behaved geometric object and named it  "imaginary"  for not having real existence. But later recieved to be a legal and one of the most celebrated members ( not only in mathematics but in applied physical sciences ) in the domain of  mathematical concepts created out of sheer necessities to solve problems. 

The expansion of dimenstions and the development of calculus were to address the intellectual curiosities,  mathematical demands and necessities to solve either mathematical problems in a precise and concise manner or to exapnad the domain of mathematics to a larger galaxy. We see the sequence of events on the developments of numbers out of necessities, from counting numbers with no zero to the one that included zero and to the set of all integers. Insolubilty of equations in integers lead to the creation of rational numbers and finally lead to the creation of the real numbers.

The developemnt of complex numbers, quaternioins and Clifford numbers, the creation of manifolds as generalizations of open regions of Euclidean spaces to consider problems over general domains - which recieved  wide space and applications in other fields of scientific studies as well, were all from necessities.

Concepts such as +, -, x, +,\, =, and all sybmols of calculus and symbols that are used in the galaxy of mathematics, were created out of sheer necessities for simplicities in representing and describibg quantities or behaviors and writing relational expressions abstractly.  

Therefore the development of mathematical concepts were results of necessities of mathematics and applied mathematics either to address problems or to exapnd mathematical knowlegde.

Dear James, 

This is a continuation of the above post! The pdf file was supposed to follow from here. 

I attached a paper from Rene on  how villagers in Ancient Ethiopia were doing mathematics (multiplication) to find the total sum of wealth a person can have from a sell of cattles for a price. The process we know  is just to multiply the unit price by the total number of cattels sold, but that technique was not available then, instead they used a different type of binary multiplication - fascinating. I have attached one file and an address : Engines of Our Ingenuity, No. 504: Ethiopian Binary Math, by John H. Liehard, www.uh.edu/engines/epi504.htm  to support the idea that mathematics was  indeed created to think abstractly and solve problems precisley.   

Dear Dejenie,

Your posts provide lots of insight and lots of food for thought about the tapestries that mathematicians have been weaving for centuries.  

In addition to zero, you will probably want to add the empty set.    A detailed study of both zero and the empty set from a contemporary perspective is given in

A. Kanamori, The empty set, the singleton and the ordered pair, The Bulletin of Symbolic Logic 9 (3), 2003:

http://math.bu.edu/people/aki/8.pdf

See pages 273-276 for the story of the empty set.    As late as 1888, Richard Dedekind excluded the empty set [Nullsystem] "for certain reasons.    Then, in 1908, included the null set [Nullmenge] 0 that contains no element at all.    And then Felix Hausdorff the empy set in his class monograph on set theory published in 1914.

@Mihai Prunescu: ...I think that the origin of mathematical concepts can be found in toys, games, puzzles.

and

@Carlos Eduardo Maldonado: ...you may know, D. Coxeter's playing as a source for all his discoveries about Coxeters sets, polytopes and many others - as Coxeter himself once told.

Evidence of the importance of playful activity can be found in

D. Whitehead, The importance of play, 2012:

http://www.importanceofplay.eu/IMG/pdf/dr_david_whitebread_-_the_importance_of_play.pdf

For example, Vyfotsky argued that play makes a crucial contribution to the development of the unique human aptitude for using various forms of symbolic representation, whereby various kinds of symbols carry specific, culturally defined meanings (p. 16).

Indeed, dear James. Math is intrinsically chracterized by a large freedom - partly because of the capacity of abstraction. You kno, working with dimension higher than 3: R4, R7, R11...

Freedom and playing are, I believe, one and the same thing. Ultimately, no human life is possible at all without freedom = playing = abstracting...

I do not have anything at all against "concrete" and "applied" knowledge. However, it makes nervous when some people only, primarily or even excludively argue about such kind of knowledge, as if despising math, play, and abstraction.

@JFP: 

Thank you very much for sharing such  a question. I always wanted to learn maths and explore biology domain. But some how, I could not. I am no one to contribute in this thread constructively. Still, may be my views could inspire some one from this field. I have always wondered that:

1. How come same genome is expressed differently in pupa as well as in butterfly. Any mathematical explanations or principles?

Mathematics and logics have (not always) had two sides, from within (internal view of its producers, creators) and from outside (external view of its consumers, appliers, users). The separation started long ago in human intellectual history all over the world (sometimes mixed with mythical thinking), and emerged slowly, but steadily.

Happily there is a fruitful cross-fertilization between the two sides, but naturally there is also a lot of activity deeply burried within the kernel which doesn't surface (yet) for those standing outside, who are just happy to see that both mathematics and logics have ever more valuable applications which are nothing but 'materialized' mathematic and logic. Examples abound over many centuries and even more so in our age, and certainly not only the physical sciences (cf. genuine mathematical subdisciplines in all of behavioral and social sciences).

It would be interesting to draw a kind of Gardner Hype Curve which shows us (tentatively of course) where on the gradual externalization and materialization of mathematical and logical concepts and techniques the different creations stand. I expect, there will be some eye-openers, surprises here. The opposite would also be very interesting: a top 20 list of unsolved scientific or technical problems which are too complex to be solved by standard engineering or other R&D.

@ James and @ Dejenie: I am very grateful for the articles you have shared here. Research Gate becomes every day a better system to use, but this is because the people using it. Thank you again; and James, thank you for most inspiring questions to discuss.

@ Ravi Sharma: Strong question about pupa and butterfly. First Idea: there were a discussion around about:

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

but also about

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

On the first sight, mathematics are supposed to be very unefective in biology. But mathematics are also very uneffective in algorithmics: we can sometimes prove that an algorithm is the best for a problem (but very seldom) but we have no mathematical method to find in general algorithms. This must be so, because it is implied by Gödel's Theorem about the fact that mathematical theories which are complex enough, such as Number Theory, must be uncomplete (if someone wants to give a clear - computable -. axiomatic system)

The point is maybe the following: the genome is like a computer program (which is a formal notion of mathematics!) and lives at a sintactic level. Caterpilar, pupa and butterfly are three situations of the working program and lives at a semantic level. Of course, every living cell of the caterpilar contains the whole genome, but different segments are activated if you have a liver cell or a skin cell, for example.

There are a lot of mathematical notions that can be found in the pupa versus butterfly algorithmic paradigm, and a lot of mathematical notions which wait in darknes to arise out of this. @Ravi, it was a good inspiration to put the question in this context.

@ Paul Hubert: Your contribution is also touching Ravi's contribution in a very subtle, hiden and deep way. I ask myself by which mechanism you have started all to speak about the same things? It is a very nice week-end.

The large caterpillar provides good amounts of blood for biochemical analysis. A collaboration with Baylor Human Genome Sequence Center at Baylor College of Medicine is using DNA collected at Kansas State University to sequence the caterpillar’s genome. “We’re very excited to help lead the sequencing of the Manduca genome, which will be extremely important for identifying genes of interest for future studies,” Kanost said. “This caterpillar is a model insect species for many projects in insect biology all over the world, especially for neurobiology. Obtaining its genome sequence will be immensely beneficial for understanding many facets of insect molecular science”(p. 7).

Dear Mihai,

Your mention of the caterpillar genome brings to mind a report on a recent study of the caterpillar genome:

Kansas State University, Perspectives, 2011:

http://www.k-state.edu/perspectives/documents/Perspectives-Summer_2011.pdf

It is observed:

Perhaps the best example of mathematical and statistical approaches adding substantially to scientific knowledge in this area comes from studies of motifs associated with recombination hotspots. The question of why some parts of the DNA sequence act as recombination hotspots and some do not has been a major focus of research attention, and remains little understood. No clear pattern was available from the 15 hotspots directly characterised from human sperm typing (p. 572).

In terms of mathematical modelling of genes, see

P. Donnelly, Modelling genes: mathematical and statistical challenges in genomics, Procedings of the ICM, 2006:

http://www.icm2006.org/proceedings/Vol_III/contents/ICM_Vol_3_29.pdf

Donnelly writes about patterns of genetic variation (p. 570).

and

Dear James,

The interplay that went between the pioneers of modern set thery regarding the concepts you mentioned, in particular, on the empty set, was interesting. The reason Dedekind did not want to use the symbol however was that he found the irrelevancy of it in his works, at the same time wanted to avoid the debates.

But in all mathematical text books and  literatures, the Russell's paradox is stated as such with out even mentioning that Zermelo independently discovered it, and not mentioned whether he used Cantor's diagonalization result as Russell did to discover the paradox.

It is an interesting read on the history behind the discovery and acceptance of these fundamental and building concepts in to modern mathematics. 

Dear Dejenie,

The following paper may interest you:

J.-H. He, et al., The importance of the empty set and noncommutative geometry in underpinning the foundations of quantum physics, Nonlinear Science Letters B, 2011:

http://www.elnaschiewatch.com/wp-content/uploads/2011/07/importance-of-the-empty-set.pdf

This paper introduces the

dimension of the empty set (p. 19),

dimension of the zero set (p. 20),

dimension of  the so-called totally empty set (p. 21).

The paper @James recommended to @Dejenie is a strong example of sincretism in the world of mathematical concepts. A big amount of mathematical concepts, which were  born independently, prove to be stronger related at a higher level. [Direction collapse here: does " higher level " always correspond to "deeper understanding", and if no, why not? However, in this particular situation, they correspond.]

The notion of empty set is in fact extremely relative. For the people living in a oasis, the desert represented the empty set. But for the people living in the desert, the desert was full of substance. On one side, this remembers me about eskimos with their 50 - 80 different words for "snow". On the other hand, about the recipe given by the french writer Boris Vian for hunting lions: """Take the desert and sieve it finely. The sand will go through, and in your sieve you will find the stones and the lions."""" [Boris Vian, "Autumn in Peking", very approximative translation by memory]

How empty is the empty set? In elementary mathematics is the absolute nothing. In set theory the empty set is something like a primitive element, used to construct ordinals, like 0, {0}, {0,{0}}, {0, {0}, {0,{0}}}, {0, {0}, {0,{0}}, {0, {0}, {0,{0}}}}, etc. In actual cosmology, there is practically no empty set, because of the abundance of black matter and black energy. The ether lives again.

Two small and partial conclusions on the way:

1.) The origin of mathematical concepts, besides toys and puzzles, cannot be separated by the origin of speech, writing and concepts in general. Like concepts in general, mathematical concepts are also dependent of the local culture and of the historical development, (development of the concept itself, but also of the society in general).

2.) There is a kind of holographic effect here. There are some (indeed very important!) concepts, with the following property: one cannot make the history of the concept, without mirroring more or less the history of the whole mathematics. Definitely, "empty set" is such an example. Also, such examples are "zero", "number", "curve", "algorithm", "equation", "decidable"... Can someone list all the "holographic" mathematical concepts? I bet that this is itself an undecidable question.

@Mihai Prunescu: The paper @James recommended to @Dejenie is a strong example of sincretism in the world of mathematical concepts.

Please write a bit about the idea of sincretism, in general.

Pointing a kind of holographic effect raises some interesting points, especially if we want to consider holographic mathematical concepts.

In general, The holographic principle is a property of string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon.

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

So perhaps you are thinking of a list of mathematical concepts on the boundary of the entire universe of mathematical concepts.   Do you agree?

@James, you are surprising me every time! Well, I was not aware about THIS definition of the holographic effect. What I was calling "holographic effect" was just the old one, known after the discovery of the holography. It means, that if someone breaks a hologram in any number of irregular parts, every part contains the whole information of the original hologram, that it contained before breaking it. This means, that one can look through the piece from different directions and one sees the object (the hologram) as like one looks through an irregular window of the same form like the broken part. [Equivalently, like the complementary of the broken part is painted in black, and the hologram continues to act as a window to the 3-dimensional holographic image.]

The definition pointed out by you is even stronger because there is a difference of dimension between the boundary and the volume... Well this was also like this in the original holographic effect, because the broken part is 2-dimensional and the holographic information is 3-dimensional. The point is, that EVERY broken part of the broken hologram contains all the hologram information. [For ideal holograms, as I suppose, it is necessary and sufficient that the broken part has a positive area.]

The metaforic sense was that holographic mathematical concepts reflect ""almost"" the whole mathematic in their historical development. Well, yes, they are connected to all other mathematical concept. They belong to the hard core of mathematics.

We have to further think about this.

@James: concerning "sincretism", it was just a missprint - unhappilly not the first or the last :-( .....  I was meaning "syncretism", a term borrowed from the history of religions. (I was using the orthography in my languange, Romanian, ..) It is how different religious doctrines and traditions go together and build new doctrines and new traditions. A good explanation of the original term is here.

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

By sincretism in mathematics I mean exactly what is happening in the article recommended by James to Dejenie, about the importance of the empty set in quantum physics. Concepts created in different contexts start to interact in a harmonic way at a level nobody has thought before. I remained strongly impressed by this example.

However, also like in the case of the "hollographic effect" cited above, I think that the "sincretic effect" is a key phenomenon when studying the evolution of mathematical concepts. At least, now I think so...

I have meant """syncretic effect""", again the same error.

Has this empty set anything to do with the mathematics of Bounias?

@ Paul Hubert: very strong contribution again. There are now five or six articles in discussion, and we need some time to make head or tail with them. However: it seems so, that this empty set has a lot to do with very creative mathematics!

I mean the following: there is only one empty set. Differences arize when we use the empty set. If person A says "let's consider the empty set" and person B says "let's consider the set of integers n > 2 such that there are integers a, b, c > 1 with a^n + b^n = c^n", both persons mean the same set. However, B puts much more non-trivial information in the context. An empty post package differs a lot from an empty noble armoir of mahagony.

It is possible that the empty set was a very good example for the whole topic discussed here (origin and growth of mathematical concepts)! And I repeat: it is a mathematical concept with strong holographic effect [touches almost all other mathematical concepts in some non-trivial context] and is very good for various kinds of mathematical syncretism [non-trivial and unexpected use and applications - as suggested also by the paper by Bounias introduced here by Paul Hubert.]

Dear Mihai and Paul Hubert Vossen,

It is a pity that M. Bounias is not RG.   I did find

http://www.inerton.kiev.ua/interest2.htm

which gives a list of publication for M. Bounias and his coauthors.   The good news is that Volodymyr Krasnoholovets (Bounias's coauthor) is on RG:

https://www.researchgate.net/profile/V_Krasnoholovets

It is definitely the case that the empty set is important in the study of both the origin and the growth of mathematical concepts.

Do you have any comments on the notion of magma in Section 4.3 (starting on page 10) of the article cited by Paul:

M. Bounias, V. Krasnoholovets, The universe from nothing: A mathematical lattice of empty sets, Int. J. of Anticipatory Computing Sciences 16, 2004, 3-24:

Bounias 2004 The Universe from Noth...hematical Lattice of Empty Sets.pdf

@Mihai:...By sincretism in mathematics I mean exactly what is happening in the article recommended by James to Dejenie, about the importance of the empty set in quantum physics. Concepts created in different contexts start to interact in a harmonic way at a level nobody has thought before. 

@Dejenie: ...The interplay that went between the pioneers of modern set thery regarding the concepts you mentioned, in particular, on the empty set, was interesting.

Mihai's suggestion that we take the syncretic approach in looking back at the origin and growth of mathematics is good.    In the wikipedia page on syncretism cited by Mihai

http://en.wikipedia.org/wiki/Syncretism#Nomenclature.2C_orthography.2C_and_etymology

the Wikipedia starts with a general definition that fits in with Mihai's suggestion:

Syncretism /ˈsɪŋkrətɪzəm/ is the combining of different, often seemingly contradictory beliefs, while melding practices of various schools of thought.

In considering the syncretism of mathematics, we definitely find an interplay of very different and sometimes conflicting ideas.   Several examples come to mind:

> infinitesimal used by Leibniz (calculus) and Nelson (nonstandard analysis) and rejected by others.

> infinity, only recently introduced by Cantor and avoided by others.    See An infinity of infinities:

http://www.google.ca/url?sa=t&rct=j&q=&esrc=s&source=web&cd=10&ved=0CFoQFjAJ&url=http%3A%2F%2Fwww.maths.ed.ac.uk%2F~jcollins%2FOpenStudies%2FMISW9.ppt&ei=aJi9U5GaEoOHyASiwYHwDg&usg=AFQjCNFMoC-M1I5P4bZvwIs5SqYYgwJDUw&bvm=bv.70138588,d.aWw

> imaginary numbers: the term "imaginary" was introduced by the French mathematician Rene Descartes:

http://www.maths.ox.ac.uk/files/imported/prospective-students/undergraduate/single-a-level/complex/complex.pdf

> complex numbers, introduced by the German mathematician Carl Gauss in 1799.

Can you suggest some other examples of discoveries in mathematics introduced out of necessity and sometimes contrary to intuition?

The beauty and elegance of mathematics is in its capacity to make people create a sound and consistent mathematical theory that guarantees the existence of an invisible object contained in an empty set ( completely defying common sense ) which is the sole cause to the formation of the visible, infinite expanse of solid and non solid galactic bodies including humans - simply the universe.

@Dejenie A. Lakew: The beauty and elegance of mathematics is in its capacity to make people create a sound and consistent mathematical theory that guarantees the existence of an invisible object contained in an empty set ...

Are you sure you want to characterise an empty set as a set containing an invisible object?   I am guessing that you meant to write that an empty set is a set containing no discernible objects (a set without content).    Such a set is analogous to an orange peel without the orange.

Do you mean, that a grand unification of physical theories is indeed possible - and it is based on ... nothing (pun intended)? I find that a fascinating idea, but like to add quickly, that we are still talking about a model of reality, a mathematical model which may be describing and explaining physical reality in an unreasonably effective way (as Witten put it) - if all experimental evidence can be catered for (wihich may turn out to be a tedious job to prove in all details).

Furthermore, if fundamental physics is wholly deterministic, as you say, it should be possible now - having a much better but not-necessarily simpler theory - to give a full account of the way that probabilities are a necessary auxiliary ingredient of former theories which missed the essential building blocks of a more encompassing and/or more fundamental physical theory.

If it can be shown how probabilities e.g. in quantum theories naturally arise by an improper transformation of the grand unification - improper in the sense that there is some sort of information loss - I guess it will be far easier to convince adepts of current mainstream theorists of the plausibility and generality of the new theory of nothing which becomes a theory of everything via a clever construction of something out of it.

Bold leaps, truly, but why! dear Volodymyr, let's give it a try ;-))

However tell us, would it have been possible for Bounias, you and a handful of other devotees (please don't take this pejoratively!) to develop such broad-sweeping mind-boggling theories and observatories while bound to a figurative-intuitive perception of the world, as you call it?

Or did I misunderstand you, i.e. do you rather mean that we shall foster both anthropological traditions including their Weltanschauungen at the same time, each to inform and profit from the other?

Perhaps this kind of dual thinking is asking too much from most people, as belief systems tend to exclude rather than include incommensurable thoughts? I don't know, I'm just wondering and pondering ...

Dear James,

Well an orange peel and the orange both contain a physical material and are not both empty. What I said was what Krasnoholovets  puts it interestingly - the fundamental theory of quantum physics and its attempt to search for the origin of the universe solely using the power of  mathematics from empty set. 

Dear Dejenie,

Sometimes the intuition underlying an analogy gets lost in translation.    An orange peel with no orange in it is analogous to an empty set with no elements in it. 

The theory that Dr. Krasnoholovets has put forward hinges on an understanding of deriving matter from nothing modelled as an empty set.    An important difference between an orange peel with no orange in it and the empty set is this: Both the interior and the boundary of an empty set have no elements.   So, if we ignore the fact that an orange peel definitely has a nonempty boundary, then, as an empty container, an orange peel is analogous to a set with no elements.

Dear Krasnoholovets, 

Great posts. But why is it the case that analytical people cannot live in peace more than few months? Then which type of people live longer in peace?

Dear James,

That is true as analogy - not mathematically though.  If Ω is the orange including it's peel and the peel is the ∂(Ω) then the peel of the peel of the orange would be empty -  ∂²(Ω) = ∅ .

Dear Dejenie,

We agree that an orange peel as an analogue of an empty set works but in fact is not a true representation of a empty set, which has neither inner nor outer content.

Dear V. Krasnoholovets,

Please tell your understanding of an empty hyperset.   Also, it would helpful if you can comment on how Bounias used the notion of an empty hyperset in his reasoning about physical phenomena.

Here is a second question.    In what sense does an empty set have inner content?

Sorry to jump in a discussion which has been underway for some time, but I feel compell to remind that there is something called philosophy, that is commonly forgotten by scientists, which can be useful.

Start with : "Math is a way by which we learn to think, whereas most sciences and desciplines just help us learn to know" . This issue was studied by the Greeks, with the distinction beween the sensitive perception of an object or a phenomenon, and its representation in the mind, whose perfect embodiement takes a mathematical form. Descartes invented the concept of physical laws, replacing the essence of things, which brought mathematical computation in physical sciences. But he was somewhat lost in his quest of first principles (which is similar to the endeavour of building physics from mathematical theorems), and was criticized about that by the empirists (Hume,...) who emphasize the role of experimentation. Kant put these matters right by telling that any theory (notably physics) come from representations,meaning concepts used to describe the properties of things and phenomena, and these concepts are further used as bricks in formal models, in particular using mathematical tools.

So philosophers show us what are the relations between knowledge and thiought. There is no experimental knwoledge which is not formated by some prior conceptualization, and there is no scientific knowledge which is not in some way realized in some experimental facts that can be checked.

In Physics, as in any field which claims to become a science, there is the need to identify concepts with respect to phenomena. Science cannot start from nothing. In the same field there can be several competing theories, and one of the job of scientists is to sort out these theories. As a consequence there is no scientific truth (a theory can always be challanged), and science does not know reality (it can at best give an efficient representation of it).

This leaves the status of mathematics :mathematics are larger than pure logic, the effeicient mathematics that we use require axioms to build the set theory or arithmetics, and we know since Gödel that then there are theorems which are true but cannot be proven. This issue is still controversial, but it seems to me that mathematics are a science. The mathematics that we know have not been built in an axiomatic way : one could do that now (the Bourbaki's way) but this is a fact that all its concepts come from the curiosity and inventivity of generations of mathematicians. Their concepts (number, algebraic structures,...) are their experimental play ground.

Thank you for this chart, James F Peters!

I am going to resonate with the post of Dutailly, in what I write below.  

What follows is less detailed than many of the wonderful and detailed posts above, as I am trying to account for what is driving the mathematical process, as opposed to specific acts or groups that have developed along the way.

I have heard (but I have not confirmed this personally) that the old greek word ``imathematicos'' translated most directly as ``one curious to know/curious to learn.''

I think this idea captures very nicely what drives the creation of mathematics by humanity; those curious to truly understand why/how something is true inevitably search for compelling answers, which leads to proofs, and etc.

In physics, people postulate rules which account for observed or predicted behaviour (predicted for ``other reasons''). Then, the mathematicians arrive and try to sort the ``truth of the matter'' by turning these rules/estimates into theorems derived from the fundamental assumptions of our world; this expresses the desire to know ``why'' something is true, given our more foundational ideas of how the universe works, as opposed to building a calculational theory of ``what will happen.''

Note that in the above, I am not concerned with whether a mathematical physicist considers their self to be a physicist or a mathematician, I am only discussing the mechanics of the process; any individual might be involved in more than one step of this process, depending on their gifts and interests, etc.

Colin,

Concerning the old Greek word, "mathematicos" (you wrote "imathematicos"), I found a reference to a work by Sextus Emiricus, Adversus Mathemticos (Against Mathematicians).

http://www.crystalinks.com/sextus.html

And apparently the word "mathematician" is derived from "mathematicos".

The best source I have found so far is

W.W. Skeat, An Etymological Dictionary of the English Language, Oxford U Press, 1882 (First Ed.), reprinted in 1983. 

Skeat confirms what you mentioned:

Mathematic (pertaining to the science of number) from Greek "mathematicos": disposed to learn (p. 366).

And you can page through this beautiful book online at

https://archive.org/details/etymologicaldict00skeauoft

See page 358 in the online edition of Skeat (the same entry appears on page 366 in my printed copy of this book).

@V. Krasnoholovets: ...the empty set (Ø) acts as the “initiator polygon”, and complementarity ( C) acts as the rule of construction.

Now I understand both how you arrive at a hyperset and where it leads, namely, a fractal object.   I imagine that the notion of a hyperset carries over into a study of subsets in real vector spaces.   For example, we can consider the boundary of a compact smooth manifold M to be possibly empty.   If V is a locally defined vector field over M.   Let U be the domain of V and assume that U is empty.  From this, we can define the index of V (over an empty domain) equal to 0.   See, for example, Sect. 1, starting on page 136, in

D.H. Gottlieb, G. Samaranayake, The index of discontiuous vector fields, New York J of Math. 1, 1995, 130-148:

https://www.math.purdue.edu/~gottlieb/Bibliography/47.pdf

Also, your explanation of the inner content of /c is helpful.

Dear Stefan,

The introduction of Victor Kraft's work on mathematics, logic and experience (Erfahrung) takes us in an interesting direction in the growth of mathematics.    It seems that the focal point is on empirical statements based on various forms of experience.   This is the point of view of the members of the Vienna Circle (Moritz Schlick, Rudolf Carnap, Frederick Waismann, Otto Neurath, Karl Menger) and the logical positivists in Germany (Hans Reichenbach, Frederick Kraus, K. Grelling and A. Herzberg).   It was Schlick in 1924 who is credited with bringing together the group of philosophers known as the Vienna Circle.    And the focus was on the formulation of empirical (synthetic) statements that are founded on experience.   For more about this in the context of the growth of mathematics, see the comments about David Hilbert, Alfred Tarski and Kurt Godel on page 15 in

S. Zecevic, Meaning and Modality by Carnap and Kripke, 2008:

https://dea.lib.unideb.hu/dea/bitstream/handle/2437/36250/PhD%20thesis%20Svetlana%20Zecevic.pdf;jsessionid=0CB4A05ADA602C29EEB73738D2EC7268?sequence=4

@jean claude Dutailly: ...The mathematics that we know have not been built in an axiomatic way : one could do that now (the Bourbaki's way) but this is a fact that all its concepts come from the curiosity and inventivity of generations of mathematicians. Their concepts (number, algebraic structures,...) are their experimental play ground.

Yes, I agree with you for the most part.   One important exception to your insightful view of the source of earlier (pre-20th century) mathematics is Euclid's geometry, entirely built on his five postulates.    By twisting and turning round Euclid's 5th (parallel) postulate, we find non-Euclidean geometry being invented by Bolyai and Riemann. 

The axiomatic approach is a handy way to build a system.   It requires quite a bit of insight to put together axioms embodying self-evident truths and  which have no obvious proofs.    Euclid and his predecessors built their geometry based on the apparent (and "guessed" structure) of the physical world.

To read mathematical origins via its fundamental notations from any time period, ancient texts must be read in context. For example, Plato's classical Greek arithmetic http://planetmath.org/platosmathematics shows a rich form of scaled rational numbers that followed 1,500 year older Egyptian arithmetic.

@Stefan Gruner: The discussion could be made more precise by looking at various conceptual "levels" at which "growth" can occur.

You have a good idea.   

In an attempt to extend Thomas Kuhn's notion of conceptional levels to mathematics, it is reasonable to consider the following scheme:

A.  structure level (your postulation level): introduce definitions, symbols, [possible] axioms, mappings on one structure into another structure, theorem (with proofs) that characterize a structure such as a vector space or topology.

B.  synthesis level (your connection level): combine structures and work on the structure level with a new combined structure.

C.  invention level (your establishment level): introduce new mathematical structures and work on the structure level with the new structures.

From this, we can construct a new level characterizing the growth of mathematics:

A + B + C

where the "+"  reads "combine".    An example of this is

S.-D. Friedman, T. Hyttinen, V. Kulikov, Generalized Descriptive Set Theory and Classification Theory, Memoirs of the Amer. Math. Soc. 230 (1081), 2014, 80 pp.

@Milo Gardner: To read mathematical origins via its fundamental notations from any time period, ancient texts must be read in context.

Yes, definitely.     Your link to Plato's mathematics is very good.   Even we tap into just a fragment of what you have pointed out would require quite a bit of effort.    Now,  for example, you have motivated a visit to the British Museum, where the Egyptian Mathematical Leather Roll has been housed since 1864.

http://planetmath.org/egyptianmathematicalleatherroll

@Milo Gardner: Here are details about the Egyptian Mathematical Leather Roll:

http://mathworld.wolfram.com/EgyptianMathematicalLeatherRoll.html

and the Rhind Papyrus:

http://mathworld.wolfram.com/RhindPapyrus.html

and the Akhmim Wooden Tablet:

http://mathworld.wolfram.com/AkhmimWoodenTablet.html

The Slab stela of the Old Kingdom Egyptian princess Neferetiabet (dated 2590-2565 BC), painted on limestone, from her tomb in Giza is in the Louvre, France (see attached image).   The hierglyphics used on the Slab stela are show in the attached image.

Dear James,

Your question is good. The following is my idea:

There are many “knowledge units” called “concepts” in our knowledge (including mathematics) mansion; they are our cognitive results through certain knowledge systems (it doesn’t matter whether primitive, simple or complicate). And, all the things in universe become the varieties of concepts, and the relationships among objective things become the varieties of relationships among concepts in our knowledge system. The relationship--rules among concepts (it doesn’t matter whether primitive, simple or complicate) are called logics. So, all the things and their relationships in universe become the varieties of concepts and logics. The “concepts-concept systems” of different forms, levels and domains with their related “logics-logics systems” form the “knowledge-knowledge systems” of different forms, levels and domains. The scientificity of the concepts decides the scientificity of the related logics: if the conceptions are wrong (unscientific), their relating logics (the relationships among conceptions) are surely wrong (unscientific). And scientificity of the “concepts-logics” decides the scientificity of the related “knowledge-knowledge system”  Thus, whenever there is “scientificity absence” defects in a certain “concepts-logics” system, there surely be the relating paradoxes in the system; or whenever there is a paradoxes in its related knowledge (including mathematics) system, this system is somewhere unscientific, so, the replacements of the “rights--wrongs” (scientific-- unscientific metabolism) in our knowledge (including mathematics) system for concepts happen and this is the growth of mathematical concepts.

Best Regards

Dear Geng,

The importance of concepts as building bricks of knowledge (meaning shared beliefs) was underlined by Kant, who stays the creator of modern epistemology. Concepts are linked together to build theories, and the link is done by formal logic (about which everybody agree since Aritotle). The key issue is then what makes special scientific theories, and how to discriminate between scientific theories ? A theory is scientifc if it can provide a falsifiable representation of reality : this is the possibility to check,by experiments, that its predictions are met that made them special. They do not tell the trruth, because they are always challenged by experiment, and can be replaced by other, more efficient, theories. On this you can read my paper on common sructures in scientific theories on this site.

Narrative rhetorics almost never probe beginning points of a culture's math system. Mathematicians at some point must point out 'ab initio' aspects of any mathematical culture. For example, we often speak of how great the Greeks were, but what were the arithmetic building blocks of its mathematics?

Plato is often misread in narrative forms that miss Greek 'ab initio' aspects of unit fraction arithmetic that later layered on geometry by the time of Heron during the Hellene period. Plato discussed his era by  http://planetmath.org/platosmathematics in a manner that scaled rational numbers by LCM m/m in a finite multiplication context. Arabs and medieval scribes as late as Galileo scaled rational numbers by LCM 1/m in a finite subtraction context. Both unit fraction systems asked algorithmic questions, answered by finite series.

Modern decimal arithmetic scaled rational numbers in an infinite series by mixing the binomial theorem with a medieval algorithm. Only after 1600 and the death of Galileo did algorithms become the core of Western Mathematics.

More interesting are the great Chinese and Mesoamerican cultures that built  arithmetic systems on modular arithmetic with different forms of the Chinese Remainder Theorem. Both used the rhythms of  astronomy as an encoding context. I'll send links to the Mayan case if anyone is interested that followed a dual use of modular and linear math in the Olmec long count.

Best Regards,

Milo Gardner

Dear jean claude,

Thank you.

1, some theories were improved or replaced because of the related paradoxes which have been produced by unscientific concepts and logics.

2, the definition of “scientificity”; I really met following troubles: the “scientificity” of conception? the “scientificity” of number conception and number forms? The “scientificity” of treating theories and techniques of the number forms（such as limit theory）? The “scientificity” of Working train of thought? The “scientificity” of scientific theories?….My studies have proved that “culture” “knowledge” “science” are confused and mixed up in some areas.

Best Regards

Number theory seems to define the 'ab initio' aspect of great ancient cultures. For example Mayan number theory links modular and linear arithmetic  forms in the long count by following the patterns established by solar and sidereal aspects of monitoring our immediate planets per:  http://planetmath.org/mayansupernumberarithmetic

@Milo Gardner: Number theory seems to define the 'ab initio' aspect of great ancient cultures.

Yes, to a large extent.   In addition, our culture is defined by our preoccupation with symbolic reasoning (consider the growth of algebra from its beginnings in the work of Omar Khayyam to its current stages, where we now have pure algebraic structures such as groupoids, rings and fields and its marriage with geometry in algebraic geometry).

http://mathworld.wolfram.com/AlgebraicGeometry.html

Our science history tells us that with its special characteristics, most concepts in mathematics are well “born”. Whenever “the growth of mathematical concepts” happens, it always means our cognitive ability improved and the evolution (small or big) of the relating theory system. However painful and difficult it is, “the growth of mathematical concepts” happens along with our human’s evolution. This is the metabolism law of universe.

To Geng,

You have a point, and the critic of the notion of "scientific concepts" has been the starting point of the deconstructionist philosophers. However it holds that these concepts, even if they can seem biased, are related to measures, physical facts. In Economics the building concepts come from accounting. Of course some would contest the concepts of added value or profit, but they have precise definitions, they are routinely implemented and used to predict or explain phenomena (such as the bankrupcy of a company). I cannot see any scientifc thought without the use of concepts linked to measurable phenomena.

This is where mathematics comes into questions. Whatever we think about the theory of sets, most of its axioms are rooted in basic observations. Except one (in the ZFC system) : one postulates the existence of a set with an infinity of elements, which cannot have a physical representant. And most, if not all, mathematics proceeds first by defining some set, and then by studying its properties. In the usual case one can prove that such sets exists, but I am not sure that it is true without the help of the axioms.

This leads to the issue of Godel"s theorem (and similar results). It shows that one can add as many axioms as we want. There is no need for these additional axioms, our mathematics are efficient enough without them. But, what if they could be handy sometimes, such has the axiom of infinity ? This is the beauty of mathematics that it can create its own objects. And the need for safeguards : the Hilbert's project is not without merits.

Sorry Wes,

but I do not see what your visual products add anything to mathematics. And anyway one of the big issue of mathematics is the concept of intinity, that is impossible to visualiize.

Wes

It is not to me to define what is a significant contrinution but to the community of mathematicians. I can say that I am one of them, no more. Your contribution does not belong to mathematics, that is a fact. Perhaps it can be useful somewhere...

The word duality is used to name different things in mathematics, its most important usage is about vector spaces, and I guess that it is not easy to visualize.

I d not want to be negative. Poincaré has written about the process of creation in mathematics and I recall (to be verified) that he said something about visuaisation.

The point that I made in my previous post what actually the contrary : in each science one creates concepts which are idealisation of some observable phenomenon. The path is observation -> definition of a concept -> measure of the phenomenon. Mathematics is the only science which has the power to create objects which cannot be physically observed (such as a set with an infinite number of elements). So one of the mystery and power of mathematics is that it escapes this process of visualisation (there are other examples than infinity, if you look at Grothendick's work you will see that he was always looking for more abstract and general objects).

Hi jean，

I think mathematical concepts are “created” and exist in their own ways different from mathematical axioms-------mathematical concepts are more fundamental than mathematical axioms; there will be no mathematical axioms without mathematical concepts. According to my studies, the defects in present infinite system disclosed by the suspended infinite related paradoxes can only be solved by the “metabolism” of present infinite concept system but not the “metabolism” of present infinite axiom system.

Sincerely yours,

OUYANG Geng

Hello Geng,

You are right : concepts precede axioms. In the set theory we have first the concept of set, then in the ZFC system we list, as axioms, the properties that must meet a set. Thus we have the same sheme as in the other sciences : concepts are precised by their properties, which are expressed in a formal system which enables us to make computations. And the fact that one can imagine different concepts make of mathematics a science, and not just a formal system. This  adds to the remark about Gödel's theorem : one could add other properties, but they are not necessary : they would not be contradictory with the existent ones, but we have an efficient enough theory to exploit the concept effciently.

If we admit the phenomenon of “the growth of mathematical concepts” (biological phenomenon? !), then the “metabolism” of mathematical concepts is a must in different scale. But when will this kind of” metabolism” happen?

@Geng Ouyang and @jean claude Dutailly, your observations about concepts and axioms are well-taken and interesting.

To move this discussion a bit further along, you may want to consider not only axioms but also hypotheses (statements with likely but perhaps not obvious proofs), conjecture (assertion believed to be provable) and claims (statements that are often fairly obvious in their contexts,  require proofs but sometimes given without proofs).

For example, consider Goldbach's conjecture (1.1), p. 6 in

M. Wang, Constructive Analysis of Partial Differential Equations, Ph.D. thesis, University of Waterloo, 1997:

http://www.math.canterbury.ac.nz/~d.bridges/theses/complete_Wang_thesis_180213.pdf

Let P(a) be a statement about some sequence $(a_n)$ and formulate a constructive theorem of the form P(a) or not P(a), p. 6.

Something more current to consider is the claim about circumcircles in

H. Edelsbrunner, Geometry and Topology of Mesh Generation, Cambridge University Press, 2001:

https://www.cs.duke.edu/~edels/Papers/2001-T-05-GeometryTopologyForMeshing.pdf

Here is the circumcircle claim:  Let S be a finite subset and in general position in the Euclidean plane and let a, b, c be three points in S.   Then abc is a Delaunay triangle if and only if the circumcircle of abc is empty (p. 4).   Edelsbrunner observes that the proof of this claim is not so obvious but it does make sense in the context of the work on Delaunay triangles.

By making such claims (without proof), one tends to nudge along the thinking and the insights about a subject.   This is also a Socratic approach to startlng a dialogue about a topic in mathematics or, for that matter, in any subject.

 “Infinite” is one of the important concepts of mathematics. The defects in “infinite” concept system have brought us a lot of troubles in philosophy and mathematics. I sincerely wish some practical defects in “infinite” concept system can be solved through our discussions.

Zeno’s Paradox of “Achilles and Tortoise” has disclosed the fundamental defects of "real infinite and potential infinite" concept system in mathematics. So we say that the unscientific (illogical) “infinite” concept system is one of the major causes of the suspended infinite related paradoxes.

In classical infinite concept system, the critical defects are unsolvable because we are not allowed to forget either "real infinite" or "potential infinite" in such paradox (exactly same situation happen in Harmonious Series Paradox). That is why this paradox family has been troubling us human for more than 2500 years.

I see the issue as the discrepancy between logic, formalised as the predicates theory, and mathematics. In many way we see logic as stronger than mathematics, because it does not rely on any ontological assumptions : indeed logic can be implemented in any field. Mathematics on the other hand needs to rely on the assumptions that the objects that it investigates do exist. Many, if not all, mathemtical theories start by defining a set (or another object) to which is assigned some properties. In some cases one can prove that such an object can exist (think to the definition of tensorial product by universal property), but in some others one cannot (such as an infinite set).

For once I would be pragmatic. We invent such objects because they are useful to develop efficient theories. So their justification cannot come from ontological proofs, but from what they can supply. Indeed the Godel's theorem show that we can add as many axioms that we want, but we do not want them because they would not have any use (as of today, but it could change). And the introduction of the efficient objects and concepts have not come from a clear, logical, construction, but the opposite way : they have been used (in the natural development of mathematics) and long after they have been incorporated in an axiomatic framework.

I am not a specialist of the field, but there are attempts to build a new set theory, using  other axioms than the usual ZFC system. They are lead by the same motivation (met notably in information theory) : to enlarge the definition of large sets. This is, in my view, what is at work in the discovery (because this is it) of new concepts in mathematics.

This is not too much disturbing because of the formal language used in mathematics, we can measure the consequences of our inventions, and circumvent any unexpected disaster.

Very good post, dear Jean. I like it, so I read and study it many times. May I discuss with people in this thread following points?

1, there are in fact many different logics in different science branches of human science system (knowledge system) and “mathematical logic” is just only one of them? So, what is logic?

2, “In many ways we see logic as stronger than mathematics, because it does not rely on any ontological assumptions: indeed logic can be implemented in any field. Mathematics on the other hand needs to rely on the assumptions that the objects that it investigates do exist.” Are the “ontological assumptions that the objects that it investigates do exist” rooted in the objective world or just our own imaginations? What is the relationship between “ontological assumptions” and concepts? Do science logics need not rely on the “ontological assumptions that the objects that it investigates do exist”?

Yes Geng, and we need to explore a bit more what is logic.

In its very basic definition, we have propositional logic, which deals with relations between propositions. So it does not consider the meaning of the propositions,it assumes only that they can take one of two values : true or false. Propositionsal logic then works along either tables truth or demonstration theory (with an inference rule). As it is, propositional logic can be implemented in any field.

In a more comprehensive vision, predicates logic consider variables (with the operators for any value, and there exist). In some ways it assumes the existence of a set in which variables exist, but without any specific property for this set (so it does not involve set theory). So predicate logic (at least of first order) can be implemented for any field.

The Godel's theorems of completness and compactness prove that predicate logic (of first order) is consistent : any true predicate can be proven in a finite number of steps, and the converse is true.

The possible extensions of propositional logic are limited: the consideration of fuzzy logic does not bring much (it relies on usual logic anyway), the consideration of infinite number of propositions is more interesting but the Godel's theorem show that it would be useless for first order predicates.

So logic can be considered as a solid and sound foudation for any rational discourse, rational being related to the consistency of the discourse, without any consideration of the  intrinsic meaning of the propositions. Any shaman can build a rational discourse about his deeds, you can be unconvinced, but the discourse stays rational.

So the key point is the distinction between rationality, whch does not consider the ontologcal value of the propositions, and science, including mathematic, which looks at the meaning (the existence, or the value supported by measures) of the propositions themselves.

Thank you, dear Jean. Might I have your ideas on following two questions?

1, there are in fact many different logics in different science branches of human science system (knowledge system). Can we have “logics of human behaviors”?

2,  What is the relationship between “ontological assumptions” and concepts? 

Sincerely yours!

Geng

What I said before - general logic or predicates logic - holds for any narrative, and encompasses any field which can be formated in false or true propositions.

But one can have more specialized logic. For instance in Quantum Mechanics some people have proposed a set of rules to axiomatize QM (see Charles Francis works on this site). But to do this one must define the propositions which will be the items upon which the logic will apply. So one needs to add concepts, and the properties which are linked with these concepts. Actually, in my view, this is not really logic, just a boolean calculus built in a theory. One could probably do the same about human behavior, but it would be a theory of its own. I think better to keep the distinction between what is logic, rationality, which addresses any field on one hand, and specialized theories, addressing specific field, even if they takes the format of propositional calculus, on the other hand..

Ontological assumptions relate to the essence of the item which is considered. It is a general word used in philosophy, but I think that, at least when one considers epistemology concept is more appropriate: it underlies that the item has to be defined, and is characterized by some properties, which can be made explicit, formalized and measured (in natural science). The traditional example of ontology is the ontological proof of the existence of God, by Thomas d'Aquin. But when one looks at it one sees that actually it relies on properties which are assigned to the concept of God (it is perfect,...).

I will add only that math is not a creation, but rather a discovery.

Thank you, dear Jean. I agree with your idea.

We can see a fact in our science history: if some concepts are not scientific the relationships between or among them are “unscientific”-------the relating logics are “unscientific” and our cognitive fruits in this field are “unscientific”.

How do you think of this phenomenon-------the close relationship between concepts and logics? No logics without concepts?

Could we have a short break for following dish?

When we study ”the meaning of zero" and the location of zero in “number spectrum” in our mathematics, an unbalanced defect can be easily discovered: “zero" appears on one side of the “number spectrum” as a kind of mathematical language telling people a situation of “ nothing, not-being,…”; but on the other side of the “number spectrum” we lack of another kind of mathematical language telling people an opposite situation to “zero”------“ something, being,…”.

We need a new number symbol with opposite meaning to zero locating at the opposite side of zero in the “number spectrum” to make up the structural incompleteness of “number spectrum” and to complete the existence of “zero”.

Sincerely yours,

Geng

Numbers are defined from the set theory, iin two different ways.

Cardinal numbers refer to the number of elements in a set, starting with the empty set (0) and (1) is the number of subssets of the empty set.

Ordinal numbers refer to the inclusion of sets in each others.

For sets with a finite number of elements cardinal and ordinal number are identical, but this becomes more interesting for infinite sets, with some issues not solved yet.

Because of the axioms of the set theory it seems difficult to invent the concept of non empty set, because even the empty set has subsets ! So the issue lies more with the different pictures of infinity.

Thank you, dear Jean,

But, when our ancestors created and defined 0 had nothing to do with set theory.

Sincerely yours,

Geng

Of course ! For centuries mathematics was built with concepts reflecting casual objects and relations. This is clear with Euclide's geometry. But the things became complicated when mathematicians tried to unify all this in a consistent fashion. And one thing that they had to give up is the idea that mathematical concepts correspond to physical objects. Now they are nothing more than abstract things, which take life through their properties.

Thank you, dear Jean,

Then what is mathematics before set theory?

Why not if our ancestors created and defined 0 had something to do with set theory, then the same thing happens on the new number symbol with opposite meaning to “zero”, too.

Sincerely yours,

Geng

As the story tells, the arabs invented the zero, but only as a marking place in writing numbers. I do not know when it enters really mathematics. Probably when notations as we know them became common, around the XVI ° century.

I have been student at the Ecole Polytechnique, which has one of the richest collection in the world of old scientific books. And I was greatly surprised how demonstrations were done in the XIX¨° century. They would not be accepted to day. Try, you will understand the weight of conventions in mathematics.

Thank you, dear Mr. jean claude Dutailly ,

ZERO as a member of “mathematic language”, it does exist in our science when we have a positional numeric system and it served as a place order-----we created and defined something in universe for our science.

Now the exactly same thing happens to a new number symbol with opposite meaning to zero locating at the opposite side of zero in the “number spectrum” to make up the structural incompleteness of “number spectrum” and to complete the meaning of “zero”.

Sincerely yours,

Geng

From the cognition point of view:

As a kind of mathematical language, the roles zero plays are decided both mathematically and linguistically.

As a basic numerical (number) element, zero locates at the right position in the Table of the Numerical Elements (Number Spectrum) such as Table of the Chemical Elements in chemistry and Light Spectrum in physics.

There are certainly many reasons for the impressive development of science in ancient Greece. But a fact is that in the same time, in the same country, democracy has been born. I think that this is no coincidence. In other civilizations there were curious minds who look at the stars or draw plans for great buildings, but for the most part these people were servants of the king, devoted to his person, employed only by him. I don't think that there was the equivalent of the agora, a place where anybody could come, expose his views, and challenge the others, however well established they were. This is quite unique in history. There is not one greek philosophy, but a long list of philosophers,with their competing schools.

It is interesting to compare the greek history to the present days Science.  Science has become a very competitive area, with great fame and financial stakes. But

assume that fierce competition has increased the pressure to innovate is a

bit optimistic. Look at the academic world, with its dominant model  (US made) of big ultra rich universities (see the Economist issue of of this week). Their teachers decide what is to be known, what is the scientifically correct thought. They decide what is to be published, through the peer review process. They decide what is new, through the chair of awards, editorial or tenur committees. Internet could have been a big progress, but look at arXiv, with an anonymous committee (comprised for 90 % of professors of the same american universities) which decide what is right to be put on the site. In any business, if the introduction of a new product was submitted to the anonymous judgment of your competitors (the peer review), there would be no innovation.The real pressure comes from outside the scientific community, when quick economic return can be expected from a new discovery. This is no surprise that Computer Sciences or Biology have made gigantic progresses, meanwhile Particle Physics is still praising a Standard Model 40 years old. Only the interest of the customers should matter, but in Science this is a very distant concern, as well as the more direct interest of students who strive to understand theories that are reputed impossible to understand. This is no longer the astrolog of the prince , but the union of astrologers who dominates the scientific world. T

Number must come before any other math topic. The second math topic is the one that the mind most often uses to solve problems. There are spatial problems so geometry is a possibility. I suggest arithmetic came before geometry and  algebra. Does not the mind easily think in arithmetic and algebra, more so than geometry?

@Milo Gardner: Does not the mind easily think in arithmetic and algebra, more so than geometry?

There is general agreement that numeracy came first for practical reasons: counting the number of things found during a hunt, for example.   This conjecture is borne out by the pictographs and rock paintings from ancient, very primitive times. 

The one thing to notice in addition to counting was the visualization of tallies and of experiences found on ancient rock paintings.    There is an implicit sense of geometry (spatial relationships of shapes, for example).   So one can also argue that numeracy and geometry grew up together in primitive cultures, before those subjects reached halls of learning.

Johann Friedrich Herbart (1776-1841)  emphasized the importance of selection and organization of content in his theories of teaching and learning.    See p. 5 in

National Focus Group, Teaching of Mathematics, 2007:

http://www.ncert.nic.in/html/pdf/schoolcurriculum/position_papers/math.pdf

These observations have a bearing on the conjecture about number or geometry coming before another math topic.    Before one can count (arrive at a tally or a number), one must first conceptualize what has been been observed spatially.   In that case, there is a strong argument that either

geometry (perception and selection of spatial structures and relationships of interest) came before counting,

or

geometry and number were arrived at in parallel (seemingly, simultaneously)

in the first discoveries of mathematical structures.

@Stefan,

Number recorded in any culture, at any time, is often hard to recognize since writing came late everywhere. For example, Mayan positional numbers recorded a complicated long count of days ( http://planetmath.org/mayanmath ) was developed by the Olmecs at an uncertain date, possibly as early as  900 BCE. The long count equivalent continuously listing of days method came late to Europe in Julian Days (JD), after positional numbers were developed by Simon Stevin in 1586 AD. The JD list was needed by astronomers to cut through the use of different calendars, the same reason that Olmecs and Mayans used the long count.

At its core, Mayan numeration was built upon base 4 and base 5 recorded as base 20, and base 18 in the long count. The base 4 aspect considered the four directions, east = ideal, south = good, west = bad, north = evil. Base 4 and base 5 was also used in a two set of knuckles 'abacus' four one hand, and five on the other, as Sanchez decoded in 1961 working in the Yucatan.

The numeration aspect of 45 California tribes recorded by Kroeber in 1920 commonly used multiple numeration system, base 4 for cosmological reasons, and another base for every day calculations. Only two California tribes (that traded items like obsidian used for knives down the coast) used base 20 for trade, and another numeration system to solve local problems. Taken as a group, the 45 California tribes were not conquered by outsiders freely used bases 2, 4, 6, 8, 10, 12, 16 and 20, and multiple combinations thereof for local tribal reasons.

In summary, in all North and Central American native cultures number came before number systems and numeral outputs. Number exists in all of their minds, and our minds before any numeration system was applied.

Is my number point clear? If not, I'd be pleased to cite Babylonian base 10 and base 60 and Egyptian base 2, base 5 and base 10 examples from which Greek chose their numeration system(s).

Best Regards,

Milo Gardner

@James,

Your suggestion that number and geometry birthed at the same time was clearly not so, per Mayans, and other cultures. It has long been known that astronomy was the first science. Chinese, Babylonians, Egyptians, Olmecs, Mayans, and many other cultures first thought in numbers, and studied the science  in numbers, writing numeration system outputs down, without geometric paradigms. Mayans and Olmecs wrote in modular and linear numeration systems, two topics not often integrated into one subject by Greeks and following Western cultures.

Best Regards,

Milo Gardner

Sacramento, CA

In present traditional finite—infinite theory system, people have been creating many new “understandings” on “infinite”, “potential infinite” and “actual infinite” since Zeno’s time 2500 years ago. But it is difficult to solve those infinite related problems produced by the fundamental defects disclosed by the infinite related paradoxes since Zeno’s time, because within the present traditional finite—infinite theory system, “the infinite related problems” are strongly interlocked together with the foundation. So, though trying very hard willing to solve “some infinite related problems” with some new “understandings” within the present traditional finite—infinite theory system, but people finally discovered that nothing can be done because “everything is perfect” in present traditional finite—infinite theory system.

I know few people agree with me, but this is true.

@Geng,

Before Zeno, Chinese astronomers developed finite numeration systems that reported modular families of planetary data sets to discovery the rhythms of the spheres. Chinese mathematicians extended math ideas and ideals to solve the Feb 1, 1961 BCE :string of pearls' alignment of the  known planets and our moon in a rigorous modular math structure called "the Chinese Remainder theorem", a methodology that reached the Greek world during the life time of Diophantus. (about 100 AD, well after Zeno). Today the Chinese Remainder theorem's (CRT) historical roots seems to have morphed to solve finite indeterminate equations without touching the infinite thoughts of Zeno.. Medieval scribes used the CRT .in the "liber abaci" in a manner that was improved upon by Gauss in 1801 "Discussions on Arithmetic" that solved the fundamental theorem of arithmetic in a manner (remainder arithmetic) that concurrently solved the fundamental theorem of algebra (that Gauss added the world of complex numbers to list finite rational roots of nth degree equations.

That is to say, mathematics grows as the problems of the world are solved in rigorous ways. Returning to Mesoamerica, Mayans were excellent mathematicians and astronomers. Mayan almanacs double and triple checked mod 13 remainders of nominal planetary cycles, such that Venus and its nominal (canonical) cycle 

584 = (236 + 90 + 250 + 8) 

was repeated five times over a period of 2920 days in a manner that Floyd Lounsbury generally extended by solving another  Diophantine equation

(4)18,960 - (41)2340 = 9100, and recorded as 1.5.5.0 in the long count.

The 9100 cycle was a correction period was  meant to compute into the future (1.413,620 days (mod 18960) = 9100 days ) with respect to  the 236 data element, in the real world. The new (to the West)  mathematical method was used three other times in the Dresden Codex, calls out to be fully decoded in terms of modern Western mathematics ( a task that I have been working on for over 20 years)..

Hence, in my view, decoding ancient math texts as originally recorded, points out the greatness of Chinese, Egyptians, Greeks. Arabs, medieval scribes like Fibonacci, Galileo and my nearby Mesoamerican Mayan mathematical friends. Decoding one problem at a time offers wonderful projects, be the problem Archimedes' square root (decoded in 2012), or a rational number problem from Egypt or the Mayan world. 

Let the ancient texts speak, as written, without modern visors worn by Zeno, Plato, or anyone else from any other time period.

Best Regards,

MIlo Gardner 

Dear Mr. Milo Gardner, very happy to hear from you again and thank you for sharing with us the creative stories in mathematical history. Our forefathers really accumulated many intellectual properties for us; it is our duty to cherish, maintain, develop such intellectual properties and pioneer more. I think we are now on the way.

Sincerer yours, Geng

@Milo Gardner

We agree that the mathematics was thriving in China, India and the Middle East long before recorded Greek mathematics.   The issue here is whether mathematics started with number theory or algebra or geometry or a combination of these or something else.

Perhaps you will agree that the very earliest mathematics had its roots in a perception of structure in nature that led to the mathematical structures in number theory, algebra and geometry.   Here are examples:

Structures discovered in China:  sequences of numbers, configurations of the planets (implicit geometry)

Structures discovered in astronomers in sourthern India: sequences of numbers, configurations of celestial bodies.

and so on.