Barry:...The limitation is not the mathematics, but our ability to measure what is going on.

I would take your comment in another direction. Mathematics does, in fact, indicate our ability to measure (grasp) what is going on. In effect, I am arguing a point of view about mathematics that runs counter to a Kantian view of the world (mathematics reflecting our ability to fabricate synthetic a priori formulations) and the Platonist view of the world where anything found in mathematics represents a discovery of the mathematics that pre-exists in the fabric of nature.

In effect, I am arguing that mathematics is usually the result of coming to terms with perceived patterns from sensed signals. The signals that we sort through (cutting out the "noise") to grasp patterns provide the threshold needed to formulate mathematics that reflects actuality.

The notion of patterns extracted by us from sensed signals comes from David Mumford, Pattern Theory Group, Brown University, as part of a paper presented at the ICM 4 or 5 years ago. If you are interested, I will post a copy of Mumford's paper.

Max Tegmak's book 'Our Mathematical Universe' is slated to come out in January, it addresses much of your question. As he puts it, 'mathematics does not reflect reality: it IS reality'.

I cannot agree with what you term here the 'central error' - I think we simply are not good enough at math, witness our appalling inability to e.g. write out the wave functions of all but the very simplest systems, or our inability to solve the pretty straightforward millennium problems. It would be tempting to believe that a different formulation of math would lead to breakthroughs, but it's kind of hard to believe since our current formulation evolved organically from expressing numbers - the very fundament of math.

Perhaps if we walked around with 4-liter brains rather than our current 1.75-odd liter ones we'd be just plain better at math ?

We never build a perfect mathematical model; a model is built for a specific purpose and its accuracy is judged with respect to that purpose

A nice statement:

“The tendency is nearly always to simulate too much detail rather than too little. Thus, one should always design the model around the questions to be answered rather than imitate the real system exactly….” ---Robert E. Shannon (1975)

Ok Mr. Ransford I will read the book, you mentioned.... Anyway last two lines of your post, rock.

Hi!

I am an electronic engineering and, therefore, I tend to see mathematics as a very powerful tool. However, even this "tool" has its limits. For example, could you bring a representation, in mathematical terms, of the human conscience?

Perhaps, with a lot of work, we could conceive a reasonable approach to some specific aspects of human conscience, but at the end, it will be just one of multiple approaches. That's exactly why we are increasingly forming interdisciplinary working groups, by integrating different areas of expertise to formulate novel solutions to specific problems.

Just an opinion....

It depends how you define human conscience, however if you consider that consciousness is emergent from 100 billion neurons then it necessarily does have an associated wave function , which of course we're totally unable to write out.

If you consider that there is more to human consciousness and indeed conscience than an emergent system (which really is easy to do - a TV show is not emergent from a TV set's complex electronics, for example) - then it still has an associated wave function, although an even more complex one.

So we hit again our sorely limited ability to do math - which led so many, beginning with Galileo Galilei, through Mario Livio et al., to say that math is God's language .....

I agree to Chris Ransford to say that math is a language (we may argue about God anyway). Considering this, that math is essentially a language, then you immediately notice the beauty and the (current) limitations of maths.

The beauty is that as a language (a very special, formal one, without exceptions, etc...) is open ended and will evolve as out notion of the world (from micro to macroscale) evolves. The limitation is that again as a language, maths can only describe conceivable ideas (concepts). But due to our "1.75-odd liter" brain (thanks HCR) together with our rather short experience in the universe, we are limited with the notions we may conceive, so is math limited.

By the way, your quotes in this post (all contributors so far) were terrific, excellent.

IMHO, I agree to Poincaré's point.

My first cent contribution is attributed to John Louis von Neumann: "If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."

While my second cent contribution, is the words of Albert Einstein: "Mathematics are well and good but nature keeps dragging us around by the nose."

Math is the only languange and way we have to describe the intricacy we observe in the real world, even if imperfect and incomplete.

@Samuel Arba Mosquera: ...Albert Einstein: "Mathematics are well and good but nature keeps dragging us around by the nose."

Apt quote. Mathematicians have made enormous strides and have made a number of significant discoveries, starting with geometry (Archimedes) and leading to orders of infinities (Cantor), algebraic topology (Krantz and others), proximity spaces (Efremovic, Smirnov), and on and on. But still nature persists in revealing its surprises and continues to baffle us. But this is a good thing. Otherwise life would be boring!

My initial submission has stem out from some sort of frustration .... .... Over the years I have seen there is a stereotype projection of modelling nature ... nothing new really ... in fact to the degree mathematics to reflect nature must start with a losing of its one-sided character and acquire new dimension which expresses dynamics, dialectical at its core. Researchers at the conference, I felt, look like sitting in an ivory-tower of sterile absolutism, unwilling to accept this dialectical nature of the real world. Particularly, mathematicians, so keen and satisfied to stay in their own world of reductionism ... I wonder where is the source of such self-satisfaction when so many things in the physical world is unfathomable!

@Arghya

There are others in the mathematical and scientific arenas who would agree with your frustration. You can find others who would agree with your point regarding the dynamic nature of reality.

A few questions:

Is it really possible to capture all aspects of 'actuality' using numbers and measurements? (Consider Lev Goldfarb's work)

If all theories and models inherently simplify that being modeled, then do we even have the tools to completely reflect actuality?

Existing on only one planet in one solar system in one galaxy of the universe, how can we expect to gain a complete perspective of the universe from our single view?

(Are we not too self-centered and arrogant believing we can capture the universe from our singular perspective?)

@Arghya Bandyopadhyay:...Particularly, mathematicians, so keen and satisfied to stay in their own world of reductionism ... I wonder where is the source of such self-satisfaction when so many things in the physical world is unfathomable!

Ahem! Instead of reduction, you may want to consider "distillation". Mathematicians discover and represent patterns that reflect actuality. Discovering a pattern and representing a pattern (in a theorem or in an axiom) is a form of distillation. By separating out the sensed signals forming a pattern from the multitude of sensed signals (with lots of noise) leads to a clearer view of nature, of actuality. So the trick is to distill from sense signals those signals that make sense and form a pattern. It is not at all the case that pattern-finding and pattern-making activities (from a mathematical perspective) leads to a reductionist view of a mathematician's view of the world.

The second part of your statement by recalling what Bertrand Russell writes in his ABC of Relativity book: It is surprising that we have done so much knowing so little. So the satisfaction problem that you mention is solved by considering what we have discovered, not what we have not discovered so far.

In his widely-read book, Mathematics: The Loss of Certainty, published in 1980, Morris Klein describes:

"Creations of the early 19th century, strange geometries and strange algebras, forced mathematicians, reluctantly and grudgingly, to realize that mathematics proper and the mathematical laws of science were not truths. They found, for example, that several differing geometries fit spatial experience equally well. All could not be truths. Apparently mathematical design was not inherent in nature, or if it was, man’s mathematics was not necessarily the account of that design. The key to reality had been lost. This realization was the first of the calamities to befall mathematics.

"The creation of these new geometries and algebras caused mathematicians to experience a shock of another nature. The conviction that they were obtaining truths had entranced them so much that they had rushed impetuously to secure these seeming truths at the cost of sound reasoning. The realization that mathematics was not a body of truths shook their confidence in what they had created, and they undertook to reexamine their creations. They were dismayed to find that the logic of mathematics was in sad shape."

Klein goes on explaining:

"By 1900 the mathematicians believed they had achieved their goal. Though they had to be content with mathematics as an approximate description of nature and many even abandoned the belief in the mathematical design of nature, they did gloat over their reconstruction of the logical structure of mathematics. But before they had finished toasting their presumed success, contradictions were discovered in the reconstructed mathematics. Commonly these contradictions were referred to as paradoxes, a euphemism that avoids facing the fact that contradictions vitiate the logic of mathematics.

"The resolution of the contradictions was undertaken almost immediately by the leading mathematicians and philosophers of the times. In effect four different approaches to mathematics were conceived, formulated, and advanced, each of which gathered many adherents. These foundational schools all attempted not only to resolve the known contradictions but to ensure that no new ones could ever arise, that is, to establish the consistency of mathematics. Other issues arose in the foundational efforts. The acceptability of some axioms and some principles of deductive logic also became bones of contention on which the several schools took differing positions."

___

I love reading the Morris Klein's book time and again, each time I get refreshed. Above, I just mentioned some portion of that book which I felt relevant to what we are discussing.

To Martin's point, we humans learn and evolve as we learn. Science, art, business, governments (hopefully), and mathematics evolve and build historically. It took about 2500 years for Euclidean geometry to be superseded and irrational numbers have been accepted for only a few hundred years, let alone 'imaginary' numbers.

Something Morris Klein may have understood is that there might not be a 'firm foundation' - in the sense of an unchanging foundation - to mathematics, logic, or science (or possibly to any human conceptualization).

No final Theory of Everything, final definitions and axioms, or Final Frontier.

There will always be a larger meta-structure of logic, numbers, and theory in which the incompleteness of one level is superseded and a push farther into the unknown made.

Looking back through history provides little evidence of any unchanging foundation - even for mathematics (let alone science, art, business, or governments).

For some, the possibility that there will be no final answers is unacceptable and makes work today as meaningless as a drop in an impossibly huge ocean. For others the possibility that things could 'bottom-out' at some unchanging level means at some point we should all just die when this is achieved (what more will there be to live for?)

These are only beliefs, since none of us can ever demonstrate anything is unchanging throughout time. I choose to believe we will always find more, deeper, more expansive concepts - so long as humanity (or beyond?) survives. I can believe in a sort-of asymptotic curve to knowledge, however we never reach the asymptotic limit point.

In this scenario, it is quite possible that what is 'mathematics' in the future will not be anything like what we have today - or that another conceptualization beyond mathematics will be found to be better at 'reflecting actuality'. I find this a challenge to find the next step or clue - and comforting that we will never run out of things yet to understand.

I think it would be instructive to read eugene wigner's paper (from the 1960's i think) "on the unreasonable effectiveness of mathematics in the natural sciences", in which he expresses some surprise that mathematics does so well at describing reality

of course, he was a physicist, and they need serious tools to do what they do, and mathematics is the cheapest form of reliable tool i know of (i am a mathematician, and i am always wondering what new mathematics will be invented to describe new phenomena in the natural world)

remember, we humans have made up all of the mathematics we use to describe reality (even if it sometimes seems as though we are discovering it), and we will continue to make up more ideas and methods, so in particular the notion that mathematics is a fixed thing is incorrect

it would be a tragedy if we were to assume that what we can describe and understand about reality is the same as reality

What if the different disciplines of science were broken out across scale - astronomy, ecology, weather, medicine, cellular biology, molecular biology, atomic physics, particle physics. And what if scale is a true aspect of reality - a true direction in space (this would mean our current 3-D model of physical space is inadequate). And what if our measurements are limited to similar-scale levels.

If this were true, then maybe the reason these disciplines have emerged along this continuum of scale is that our mathematical tools for measuring (most anything) have limitations and cannot cross any significant distance of scale. This could mean our pervasive use of positional positive base systems (usually the decimal system) has limitations of scale which affect what we can measure, what calculations we can make, and therefore what mathematical theories we can devise.

This might be an area for significant advancement in mathematics and in science. However it might take some re-structuring of our thoughts to engage - even of our concept of reality to include scale. We require new tools which can cross scale - large 'distances' of scale.

A potential direction: The 400 year old (from an applied perspective, 1500 years old since its invention) use of decimals, representing Real numbers, needs to be advanced with a numeric system that can encompass complex numbers. To accomplish this we will need to expand into an undefined area of mathematics - that of negative base exponents and logs. If we can invent a negative-base system, then we can easily produce a singular numeric representation of a complex number with, say, a base of -2 and an exponent of 1/2 [eg. sqrt(-2)]. This provides the means for representing complex numbers as complete singular values - rather than pairs of decimals involving an undefined 'i' term.

If, in addition, this numeric system also expands the operations used to build it (note that decimals use addition, multiplication, and exponents - which are each reversible operations with subtraction, division, logs) - say through use of base 'e' and including integration/differentiation into the structure of a number - then our numeric system will have abilities completely un-imagined using our current tools. The integration/differentiation operations might even cross scale (using the natural logarithm 'e' allows for indefinite recursion for integration-differentiation, possibly as positional operations)

The above situation might just keep us all occupied for the next few hundred years working it all out.

@Donald Palmer:...What if the different disciplines of science were broken out across scale - astronomy, ecology, weather, medicine, cellular biology, molecular biology, atomic physics, particle physics. And what if scale is a true aspect of reality - a true direction in space (this would mean our current 3-D model of physical space is inadequate). And what if our measurements are limited to similar-scale levels.

Raising the question about different scales across various sciences suggests a very different direction from the one suggested by Martin Kovar. Instead of looking for various scales, it makes more sense to characterise each science in terms of the search for structures in nature. So, for example, a chemist would be looking for molecular structure; an astronomer, galaxy structure; a physicist, spatio-temporal structures; and so on. The secret to understanding science and mathematics boils down to learning what structures have been found or what structures might be found or what refinements are possible with those structures currently being investigated.

As I like to tell others, if you cannot fathom somebody about Mathematics, study more Mathematics and it should be clear. If it isn't clear enough yet, you may have made a new contribution. Even mathematicians admit when somebody doesn't make sense. Why? Its level of rigour is fundamental. When we ask ourselves about anything in the technological age, the vast majority of it all boils down to modelling and mathematics. It doesn't have to relate to geometry, it could be as simple as why your smartphone is able to carry out "multi-tasking" or what process says what order they are done, and many try to prove mathematically how well these processes work.

The limitation is not the mathematics, but our ability to measure what is going on. This leads to uncertainty in the data that is fed into the mathematics being used, and hence a limitation on how far maths can take us. This is not to say that there no need for research in maths - to the contrary, this is a vital part of making progress. Every time we make an advance in our ability to measure things, we find we need to revise our models, and potentially extend the maths being used to capture new processes.

This is not to say that maths can solve everything. Some things will always be beyond our ability to determine (consider the implications from Heisenberg's uncertainty principle for example). Also, stochastic processes can be describe by maths, but only in terms of their impact on overall distributions.

Barry:...The limitation is not the mathematics, but our ability to measure what is going on.

I would take your comment in another direction. Mathematics does, in fact, indicate our ability to measure (grasp) what is going on. In effect, I am arguing a point of view about mathematics that runs counter to a Kantian view of the world (mathematics reflecting our ability to fabricate synthetic a priori formulations) and the Platonist view of the world where anything found in mathematics represents a discovery of the mathematics that pre-exists in the fabric of nature.

In effect, I am arguing that mathematics is usually the result of coming to terms with perceived patterns from sensed signals. The signals that we sort through (cutting out the "noise") to grasp patterns provide the threshold needed to formulate mathematics that reflects actuality.

The notion of patterns extracted by us from sensed signals comes from David Mumford, Pattern Theory Group, Brown University, as part of a paper presented at the ICM 4 or 5 years ago. If you are interested, I will post a copy of Mumford's paper.

G'day James,

I would be interested in reading that paper. I agree with you - my focus is on building models based on the information we can extract from observations. There are a number of tools that have come from mathematics that help with this. Uncertainty in the observations will limit how far mathematics can take us, but without mathematics, we would not get as far as we have. All depends on the magnitude and nature of the uncertainty. Thus we need to acknowledge there is uncertainty in the data, and use our understanding of that uncertainty, coupled with mathematics to help extract information from the data.

Good evening, Barry,

from Winnipeg at sunset time! Many thanks for your comments.

The attached pdf file contains the paper by David Mumford.

Best regards,

Jim

P.S. Please let me know any reflections you have about the paper. It is a gem.

Math is an unambiguous way to model reality - it approximates but in most cases does not reflect actuality but rather an ideal version of it.

There are the mathematical equations and measurements we make of reality and there are the interpretations we make of those equations and measurements (ie. our theories). Understanding reality is not a mathematical endeavor, in that the equations can be equally good for multiple interpretations. It is the interpretations and theories of what is going on that constitute science and understanding. The mathematical aspects, by themselves, do not provide this.

Newton's equations of gravity did not suddenly become 'wrong' when Einstein put forward General Relativity. Our interpretations of what the equations meant changed.

The equations of quantum mechanics are very accurate - but ask a physicist what is really going on and you might find our understanding lacking.

I think people are expecting too much of mathematics and not enough of human understanding. If all we have are equations and (even very accurate) measurements - do we really understand what is going on?

"It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it. " - These are the famous few lines due to Riemann. My initial submission has never intended to pose any doubt such as "Is Math really important?" But rather be open to revision and extension, which I find at the low ebb among people of Mathematics. This is my observation, I wish I am wrong.

Arghya,

Your observations are extraordinary. If I understand you correctly, you are saying that there is a low ebb among the mathematicians that you know, when it comes to revision and extension.

The idea of revision in mathematics is very difficult. Most of the mathematics that I know is flawless. The idea of revising mathematics does not come into the picture.

However, if by "revision" you mean "clarification", then it is common for one mathematician to clarify bits of mathematics from other mathematicians.

The idea of extension of mathematics is more common. Those who are active in mathematics are ever and again extending parts of the mathematics that are in the research area for a mathematician. I find that mathematicians are very loyal to other mathematicians and gently and subtly extend the earlier theories. The landscape provided by mathematics is a lot like life a garden, which is in a constant state of growth and renewal.

Mathematics is a nice garden to get to know and spend some time in.

Its a very Good Morning James!

Indians are naturally gifted in Mathematics over perhaps thousands of years, even the common men in the streets don't use calculator to do common arithmetic which I found so common in US and in Europe, & felt it miserably "extraordinary". Now comes the question of mathematicians, I prefer to reserve the word mathematician and rather intend to say "people of mathematics" for some obvious reasons which I believe needless to mention. Lastly "revision" has nothing to do with whether mathematics is flawless and so etc .... revision and extension are so closely connected and let me tell you one thing I always keep a little apprehensive with however little research I do, never so sure whether I am flawless or not. I may be suffering from psychological problem :-)

Arghya,

What a beautiful realization! I have the similar feelings too. To me it comes to a point that as long as we are in a finite domain of existence - we are good to understand everything with our so called invented mathematical logic. The reality is that we are totally stupid when we start thinking that our projected frame of reference does not work in infinite domain. In other words we can't draw our frame of reference on which my logic is based, in infinite domain. Powerful and proud scientists think if they proceed with whatever Science they have one day they unfold the infinity. Science, is in this world since civilization starts, scientists finds many many things, but now today's invention to infinity is infinitely many steps to go. So, even human can do many things but it is infinity steps to go to understand the Universe/ NATURE.

Only way, I find that we need a separate and/or alternate path to realize the Reality. I find that if we try to understand what Swami Vivekananda said about Adaitya-bad may have some answer it it. As a first step I read those articles related to:

Quantum Physics, Consciousness and Hinduism. I also follow what Physicists Dr. Amit Goswami said about consciousness and Quantum measure. I am still in dilemma, but trying hard to understand it in alternate way.

Regards.

Asit

Dear Asitda,

Nice to get a comment from you. I hope If you have seen the ongoing discussion with my initial and subsequent submissions, also others view. Researchgate is such a good forum. Every opinion is equally important. You have rightly pointed out and found the chord from which I am saying all these, our own philosophy has taught us how to be humble in search of truth, say it be maths or anything else. And that my research, my mathematics may not be flawless (although so many 'so-called' mathematicians have put their seal of authenticity on it) keeps me haunting. And this to realize, I am really bound to say, you should have a connection with our philosophy otherwise I'm afraid it may be misinterpreted.

Thank you Arghya. I have seen your profile in ResearchGate and find that you had graduated from Jadavpur University - I was also from Jadavpur University- did my Masters & PhD in 1984 and 1990.

I appreciate your question and answers - In my 30 years of academic life - I find only one person who gives the idea of alternate Science. His name is Prof. P. C. Das Chowdhuri, leaves in Golpark, he did his DSc from University of Paris, ex professor of Indian Statistical Institute and was a student of S. N. Bose.

If you find him somewhere please talk to him and his research - may be it will help you to achieve your objective "To Understand Absolute Truth".

Regards.

Asitda

@Asitda ... thank you for your response. I will certainly try to find Prof. P. C. Das Chowdhuri and if I get the opportunity to talk with him I will seek his blessings.

Arghya and Asit,

I wish I had been born in India to gain insight in this very interesting discussion about revision and extension of mathematics. Instead, after reading Ghandi's Experiments with Truth when I was an undergrad, I gained a deep appreciation for little living things (microbes) that we sometimes step on when we walk and a reverence for life.

I believe I understand your points of view about humility in doing mathematics. Going back to the issue of flawlessness in mathematics--an important issue, it is my guess that most mathematicians do not think their results in mathematics are flawless. It is just that a finished work is often strong enough to provide a stepping-stone to the next stage in the evolution of mathematics.

Examples of this evolution can be found in the progression

--from Euclid to Bolyai, from Euclidean geometry to non-Euclidean geometry,

--from Leibniz and Newton to Cauchy, from infinitessimals to limits,

--from Hausdorff to Krantz and others, from set theoretic topology to algebraic topology,

--from Efremovic to Leader, Naimpally and others, proximity spaces

Dear James.... you are a senior Professor of Mathematics with huge research experience in different topics. Your comments and remarks are important, to say the least, for me.

Arghya,

Many thanks for your post. And the question for this thread is excellent.

We (all of us) are are working towards an understanding of mathematics having the strength to reflect actuality. Actuality itself is elusive and defies description. It is by no means easy to harness nature with our mathematical representations. Personally, I think Efremovic comes closest to providing an approach that almost perfectly reflects actuality. His approach is a gentle one. The basic idea is to view the world in terms of those things that are close to each given thing (doing that leads to a uniform topology).

Dear James, ....can you provide some more leads to the work you mentioned? although I know very little of topology, .and frankly don't know how far I will be able to understand but would like to ...

Arghya,

I am glad that you are interesting in going further with the ideas in my previous post.

It is Efremovic that holds the key to the approach I mentioned. Efremovic published a paper (in Russian and so far not translated) on infinitessimal geometry during the first part of the 1930s. And Smirnov followed up with a paper that clearly explains Efremovic's notion of proximity space (and infinitessimal geometry). Smirnov published his paper (in Russian) in 1952 and that paper was translated in 1964 by the American Mathematical Society. It that paper that I have attached to this message.

@ Prof. James.... ....Thank you really.

Dear James & Agrhya,

I have problem to download the paper James posted. Is it possible to send me a copy by e-mail. My email address is: [email protected]

Asit,

I just now sent to you the pdf file containing the paper by Smirnov. I hope you find it interesting. Any comments will be welcome.

See, also, my RG page, where I have a number of papers about Efremovic's approach to proximity spaces.

I agree with dear @James that the limitation is not in mathematics, but in our ability to measure what is going on. Mathematics did solve almost all issues that were unsolved, but in the same time, these solutions open many new open problems and interesting issues. This is the beauty of Mathematics. Without Mathematics there is no beauty in life.

Dear Dr. Arghya Bandyopadhyay,

Mathematics has potential to reflect reality. If this is not true how we model, design and analyze the actual system.

@Abedallah M Rababah:...Without Mathematics there is no beauty in life.

I agree. The universality of mathematics in the world view of all of us can be gauged by the fact that each person perceives structures in the world. The perception of structure is at the threshold of mathematics. And the perception of structure is also at the threshold of the beauty of life.

Well said dear @James, "The perception of structure is at the threshold of mathematics. And the perception of structure is also at the threshold of the beauty of life."

@James & @Adedallah

I hope you are not discounting the evolutionary aspects of mathematics in your conception of the "universality of mathematics".

There are many aspects of mathematics which have limitations and short-comings, which, when applied, can influence and even mis-direct studies in physics.

The law of large numbers, for probability and statistics, presumes an averaging of some characteristic of objects being measured. So use of these mathematical tools immediately brings an unknown (the 'averaged' characteristic) into the picture.

While there might be some 'ultimate' mathematics that is universal, the mathematics we use today has many constraints and limitations and is not so universal.

Quoting from the original Question:

"Is our passion for mathematics disabling us to look beyond to embrace some new (mathematics, perhaps)?"

One might say that success, however small, invites us to do more of the same. It's built into our genes and applies equally well to our experiences with mathematics.

For instance, infinitesimal numbers once were quite successful in differential calculus and in analyzing physical phenomena. Infinitesimals were actually based on nothing ("spirits of deceased numbers"). They were gradually abandoned when Cauchy gave a proper definition of limit and continuity. (Note that, in the 1960's, infinitesimals got a second life with a logic method of non-standard models.)

History confirms that mathematicians do not remain blinded by topics or methods. From Euclid up to the nineteenth century, mathematicians got stuck on the parallel postulate. It was a novel idea that broke the spell: rather than defining "points", "lines", etc. (as Euclid attempted), axioms and logic are the *only* source of information on undefined "subjects", inviting to search for other models than the intended one. So the undefined "straight lines" could become great circles on a sphere or (more generally) geodesics on a Riemannian surface.

Another famous example began with the solution of quadratic equations in Babylonian times. Then came cubic equations (16th century) followed rapidly by quartic equations. It took about two centuries to change sides and realize that the quintic has no general solution; this resulted in a beautiful, useful, and strong Galois Theory.

In each of the above examples, mathematicians looked beyond the obstruction. Moreover, the new mathematics that developed to escape the dead end can hardly be seen as the result of "coming to terms with perceived patterns from sensed signals". I am referring here to a remark of James that is only partially true. Important parts of mathematics indeed serve the needs of physics. There is an equally important part of mathematics that just serves the needs of mathematics itself. That mathematics cannot provide for all its own needs is demonstrated by Godel's first incompleteness theorem. It is thinkable that physics, too, may one day show up with truly unsolvable mathematical problems -- not a "human disability to look beyond some obstruction".

@Donald G Palmer: I hope you are not discounting the evolutionary aspects of mathematics in your conception of the "universality of mathematics".

Perhaps you will agree that the term "growing (enriching)" is better suited than the term "evolutionary" in describing the changing character of mathematics. It is not so much that the mathematics of old (e.g., the geometry of Euclid, differential geometry of Riemann, infinite-dimensional vector spaces, topological spaces from Hausdorff and the more recent algebraic topology from Krantz and others) has changed but rather has been enriched with new relations and new theorems.

In several threads discussing the nature or the possibilities of mathematics, I notice some ambiguity on what exactly is being covered by the word "mathematics". Here is a rough sketch of three quite different "objective viewpoints".

1. Human activities/thoughts as communicated in a human network;

2. A collection of concepts and results;

3. A method to derive truths from basic hypotheses.

Also, mathematics has changed in time (from either viewpoint). Donald said: "the mathematics we use today has many constraints and limitations and is not so universal". That is about as true as saying that a building, which is being renovated, has limited comfort and is not so well-suited for inhabiting.

In viewpoint 3, mathematics is close to universal. In viewpoint 2, we truly have a building under renovation. In viewpoint 1, mathematics is basically a human contrivance.