A mathematician belives nothing until it is proven

A physicist believes everything until it is proven wrong

A chemist doesn't care

biologist doesn't understand the question.

--------------------------

Biologists think they are biochemists,

Biochemists think they are Physical Chemists,

Physical Chemists think they are Physicists,

Physicists think they are Gods,

And God thinks he is a Mathematician.

Dear Mikko!

Generally, scientific research supposes some analytics about observed data or events. Now mathematics is rather improved and allows to use different levels of rigorness. If it is possible, the strict dependence laws are the best that researcher can find. Usually these laws use differential equations to describe dependences that can be observed.

If the observed dependences look like too complicated to find strict laws, then researchers use tools of probability theory and stochastic processes to describe what they observed.

In the very rough and basic case, when researcher have deal with new kind of data, he/she uses statistical tools to describe what he/she observed.

Nowadays, i think, nobody publish raw data as a scientific result without any basic explanation of it . So mathematics offers a lot of tools for researcher, for any level of detalization of dependence laws description.

Interesting question. I agree that the rigor of mathematics would be very beneficial to other scientific disciplines. However I'm not sure that the methods used in its "purest" form can be applied to science in general. This is mostly because "pure math" is not falsifiable: given some axioms, you can deduce all the other properties, and you can PROVE that all these properties are correct. Science in general (and experimental science in particular) instead it is falsifiable: in other words, you can never be 100% that some theory is true, you can only say that that theory is in accordance to the experiments that have done so far. That's why being a scientist is frustrating. The only thing that you can really prove is that your theory is wrong! (or keep confirming it with the fear that someone will prove it wrong in the future!)

It is intriguing that the falsifiabity of scientific theories then has effects on applied math (but not on pure math). To cite A. Einstein:

"as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

Thanks for the answers! Davide is getting close to what I had in mind when asking the question.

To narrow down the topic, I would be interested in hearing what value the rigorous work of mathematicians has in other fields – particularly those operating in the language of mathematics. Physicists are used to "deriving" laws by computations that are often mathematically unjustified but lead to verifiable predictions. Mathematicians, on the other hand, are still trying to "prove" say Fourier's law -- essentially that in a steel rod heat flows from the hot end to the cold one -- a well-known phenomenon that no sensible mathematician would ever dispute.

The question is, then, whether a venture of the latter type can genuinely serve other fields, or whether it is limited to satisfying the curiosity of the oddly pedantic mathematician.

Mathematical rigor is something what drives science, albeit in two opposite directions. Once you have a correct equations describing something interesting you may predict how it (i.e. the described phenomenon) will behave in other, often extremal and perhaps not yet available conditions. Sometimes you succeed (and get the Nobel Prize), sometimes your rigorous description fails. Just because it is so precise but lacks some important ingredient. Nevertheless, even in such unfortunate situation you have a very good reason to continue your scientific work.

In answering the question above and especially in relation to physics,the role of mathematical rigor is twofold:

1-. First, there is the general question, sometimes called Wigner's Puzzle, "how it is that mathematics, quite generally, is applicable to nature?" However, scientists assume its successful application to nature justifies the study of mathematical physics.

The Unreasonable Effectiveness of Mathematics in the Natural Sciences is the title of an article published in 1960 by the physicist Eugene Wigner. In the paper, Wigner observed that the mathematical structure of a physics theory often points the way to further advances in that theory and even to empirical predictions, and argued that this is not just a coincidence and therefore must reflect some larger and deeper truth about both mathematics and physics. Dirac equation is a very good example. Dirac predicted the existence of antimatter few years before it was discovered.

2-Second, there is the question regarding the role and status of mathematically rigorous results and relations. This question is particularly vexing in relation to quantum field theory.

Both aspects of mathematical rigor in physics have attracted considerable attention in philosophy of science.

Of course, the math. rigor, or discipline, is very useful for the minds of all scientists, in the same way as physical exercises are useful for our bodies: we need to be able to track the exact consequences of our claims, including various calculations.

However, IMO, probably, the most important, strategic, gift of math. to science are various formal structures that it offered to science (mainly to physics)---e.g. vector space, group, topological space, manifold, Riemannian space. This was also the Bourbaki's view as well.

Moreover, even in the negative sense, when one realizes that there are no satisfactory formal structures available and we have to seek something *completely* new---as the situation we are facing now---we will be doing this based on our previous *structural* experience.

In scientific phenomena mathematical modeling is very important to prove it and to justify the results.Twentyfive years back, to study the magnetizing inrush current pattern on CT secondary, for the development of "one cycle relay" for transformer protection was possible due to mathematics by me(see web B.T.Desai)also developed a fast acting analog filter for getting sequence component from regular load current signal and to find responce.

Negatively

David Gurarie is quite wrong. Mathematical rigour was discovered by the Greeks and recognised as a philosophical answer to the capriciousness of their gods. Socrates was consequently condemned as an atheist "corrupting the young". For us, the rigour learned from Greek mathematics (and the immediate stimulus was Aristotle's philosophy) was applied to theology. Aquinas in his "Summa Theologica" specifically showed how Aristotle had to be corrected to be read consistently by Christians. The crucial word in the last sentence is "consistently". In theology, rigour was considered to be of the essence.

The mediaeval Christian scholars, strongly influenced by the then brilliant Islamic scholarship, built on this foundation of consistency to lay the groundwork that Galileo, Kepler and others relied on. "Laws of Nature" is an idea actually introduced explicitly by Descartes: the God of the monotheists is emphatically not capricious!

Lev Goldfarb is correct in my opinion to point to the fundamentally philosophical nature of rigorous thinking. He even phrased his comment in terms Aristotle would have approved! But I think it is insufficiently recognised today that the scientific enterprise itself was underpinned and encouraged a thousand years ago and subsequently by Christian reflection on the reliability of God.

Mikko Stenlund is driving at this wider philosophical context when he refines his question to ask "whether [such] a venture ... [must be] limited to satisfying the curiosity of the oddly pedantic mathematician". My answer to this is that we do science because we recognise and seek truth; the love of truth is not itself strictly scientific but is the philosophical context for scientific activity.

Mathematics (modern day and partly Greek) is not a science, but a highly formalized language, based on overzealous pursuit of 'rigor' and formalism.

Real Sciences (physical, social et al) deal with Real World and its phenomena, Mathematics lives in a world of abstractions (their "spaces" are little more than verbal constructions).

Could a pursuit of "formal language" be useful for real sciences? It depends on how one does it (uses or abuses).

Whatever role math. played in the growth of modern science, its own development was primarily motivated by scientific questions (from Archimedes to Galileo, Newton and on). Indeed, most great mathematicians of XVII-XIX century were first and foremost scientists, knowledgeable and motivated by science.

It all has changed in the XX century, when "abstract mathematicians" detached from anything "real", and often prided themselves in pursuit of abstraction, like infamous Bourbaki (after 60 volumes of abominable "formalism" they didn't even come to foundations of mechanics).

XX century math. contributed minuscule amount to the science, in striking contrast to revolutionary inputs of scientific computing.

David, talking about "rigor", you are not accurate in so many ways that I don't know where to begin. ;--)

Even before you provided this extended answer, I new that your short negative answer was not quite addressing the original Mikko's question, but is a quite understandable reaction to the hardheadedness of many modern mathematicians.

However, one should not confuse the poor general scientific education of many modern mathematicians---which applies equally to the professionals working in all other disciplines---with the critical role the math. language has played in the development of science. Science is not thinkable without some kind of formal language: spoken languages are not suited for that purpose.

Bu the way, Bourbaki wrote 27 books and not 60

( http://www.bourbaki.ens.fr/Ouvrages.html ).

Finally, the "revolutionary inputs of scientific computing" are, first of all, not revolutionary at all, and second, they would have been impossible without the "math. rigor". ;--)

All sciences are based on mathematical models. The notion of proof belong to mathematics alone. Science is concerned with inventing relevant mathematical models (map) for specific phenomenal domains. Science is also concerned with the verification of the coherence of these mathematical models within these specific phenomenal domains.

Learning mathematics is a training in rigor. How can someone can imagine a appropriate model in his/her domain without having the tool box of model making which provide mathematics?

Mark Twain said : "When someone has only a hammer in his tool box, then the every problem look like a nail."

I think that David Gurarie, in correctly characterising science as the study of observable reality and maths as the formal study of abstractions, is incorrect in thinking that Reality is supreme, with his consequent assertion that the development of maths was driven by science. He forgets what a slippery concept "reality" is.

First of all, he is quite wrong to suggest that Greek maths was driven by science. He cites Archimedes, but Archimedes was interested in a variety of arcane matters, such as counting the grains of sand in a Universe big enough for stellar parallax to be normally unobservable (for which he invented a sort of logarithmic notation), or calculating the exact volume of an ideal cone (for which he effectively invented calculus, using a method of infinitesimals and arguably influencing Newton). Remember, although we can plausibly interpret Greek astronomy as being the "scientific" study of the stars and planets, the Greeks were interested in the stars for religious and philosophical reasons. It was precisely their other-worldliness that attracted them! Actually, Aristotle formalised this by developing the distinction between the celestial and the mundane.

The Greeks were fascinated by such problems as squaring the circle or doubling the cube - these are entirely abstract! Apollonius' deeply esoteric work on "Conic Sections" (see http://www.math.rutgers.edu/~cherlin/History/Papers1999/schmarge.html ) was only recognised as being of any practical value over a millennium later, by Kepler. Examples can easily be multiplied of mathematical concepts obtained entirely from formal consideration of abstract concepts with no possible use conceivable to the authors. The development of imaginary numbers springs to mind.

Secondly, and perhaps more importantly, to speak of the "Reality" of science and (pejoratively) of the deeply abstract "verbal constructions" of mathematicians is to reduce reality to what can be seen. "The wind bloweth where it listeth" said Jesus (John 3:8); can we see the wind? The poets do not speak of what can be seen with the physical eyes. The tautologies discovered by mathematicians are just as real as the chair you are sitting on. In fact, we only understand what our eyes tell us about the external reality of the world because we can speak about it, and there is a huge literature on the psychology of perception. We cannot see or grasp anything without the underlying apparatus of abstract thought.

Here is a question. In the previous sentence, are "seeing" and "grasping" metaphors or not?

Science is the natural and the fact. while mathematic is the sign to explain the natural science.

How does mathematical rigor influence Science?

This is one way how:

"If there is something very slightly wrong in our definition of the theories, then the full mathematical rigor may convert these errors into ridiculous conclusions."

— Richard P. Feynman

(11 May 1918 - 15 Feb 1988)

American theoretical physicist who was probably the most brilliant, influential, and iconoclastic figure in his field. His lifelong interest was in subatomic physics. In 1965, he shared the Nobel Prize in Physics for his work in quantum electrodynamics.

Feynman is of course correct, and he points up the great usefulness of full rigour: even small errors can be highlighted! But more usually it is subtle errors, such as the underlying assumptions of both Euclid and Newton that their axioms were self-evidently true where actually they were contingent (in Euclid's case) or actually false (for Newton's assumption of the existence of absolute frames).

An interesting modern example of this is the explanation of the massive scalar "Higgs" boson. The possibility of this was thought to be excluded by the Goldstone theorem (Goldstone, Salam, Weinberg, Phys.Rev.127, 1962, 965) which seemed to prove that spontaneous symmetry breaking entailed zero-mass bosons. Higgs' seminal paper (Higgs, Phys.Rev.Lett. 13, 1964, 508) was predicated on the recognition that "theories with a local gauge invariance fail to satisfy one of the axioms (manifest Lorentz covariance) on which the 1962 proof of the Goldstone theorem depends." (Higgs, Comptes Rendus Physique, 8, 2007, 970).

Rigor is clarity of concepts and precision of arguments. Therefore in the end there is no question that we need rigor. I have already given an example, Dirac Equation, as a good example where rigor help to advance progress of physics. Unfortunately, there are cases where actually insisting on rigor delayed progress in physics. One such example is in statistical mechanics:

The lack of rigorous proof of Boltzmann ergodicity delayed the acceptance of the idea of statistical equilibrium. The rigorous arguments were faulty--- for example, it is easy to prove that there are no phase transitions in finite volume (since the Boltzmann distribution is analytic), so this was considered a strike against Boltzmann theory, since we see phase transitions. Since there was no proof that fields would come to thermal equilibrium, some people believed that black-body light was not thermal. This delayed acceptance of Planck's theory, and Einstein's. Statistical mechanics was not fully accepted until Onsager's Ising model solution in 1944.

Issam: many other examples can be added. I am thinking of the development of analysis (calculus treated rigorously), or even Galileo's mathematics which systematically avoided the extensive arithmetic that Kepler used on the grounds that ratios were rigorous, but the correctness of the arithmetical operations was at that time debatable.

The question is, is it fair to say that "progress in physics was delayed"? Or should one say rather (and much less pejoratively) that the development of a clear understanding of the physical models took time? I think that understanding cannot be rushed, and that the demand for rigour encourages a clarity in thinking which must surely be a Good Thing.

Chris: Notice that I started my answer with"Rigor is clarity of concepts and precision of arguments. Therefore in the end there is no question that we need rigor"

The case of statistical mechanics is, I believe, a good example of how rigor delayed progress in physics simply because the theory was correct.

however, what if the theory was not correct? The answer is we need rigor always.

You are absolutely right in saying "I think that understanding cannot be rushed, and that the demand for rigour encourages a clarity in thinking which must surely be a Good Thing."

Rigor is part of the advancement of physics and not necessarily a delaying factor.

I thank you for pointing this out.

Reality is the greatest abstraction of all. 2+2=4 is more real than String Theory (which will never be observable due to high energies involved) as we all know when we are paying our taxes.

On the relationship between mathematics and science -Part 1:

The two big breakthroughs in physics in the 20th century owed much to mathematics. The first was the formulation of quantum theory in the 1920s, of which Dirac was one of the great pioneers. The theory tells us that, on the atomic scale, nature is intrinsically fuzzy. Nonetheless, atoms behave in precise mathematical ways when they emit and absorb light, or link together to make molecules.

The other was Einstein's general relativity. More than 200 years earlier, Isaac Newton showed that the force that makes apples fall is the same as the gravity that holds planets in their orbits. Newton's mathematics is good enough to fly rockets into space and steer probes around planets, but Einstein transcended Newton. His general theory of relativity could cope with very high speeds and strong gravity, offering deeper insight into gravity's nature.

Yet despite his deep physical insights, Einstein was not a top-rate mathematician. The language needed for the great conceptual advances of 20th-century physics was already in place and Einstein was lucky that the geometrical concepts he needed had already been developed by German mathematician Bernhard Riemann a century earlier. The cohort of young quantum theorists led by Erwin Schrödinger, Werner Heisenberg and Dirac were similarly fortunate in being able to apply ready-made mathematics.

An interesting qustion which may not possess a definitive answer. Nowadays mathematics and science are strictly separate areas of study, at least when it comes to pure mathematics not so applied mathematics or statistics. Historically mathematics and the sciences were joined in natural philsophy until maybe the 19th century after which they became truly separate which eventually also led to a great dustinction between the arts and sciences.

It should be noted that to some degree the laws and principles of physics, chemistry and biology were written down as statements about phenomenon before they were ever written in terms of equations. This does not mean to say that mathematical thinking did not influence the eventual stating of these principles. However, in modern times mathematics has come to be more of a tool in science in order to induce rigor in arguments leading from these principles/laws.

Rigor works in two ways, firstly its useful in making arguments which hold together and secondly it allows the construction of symbolic representations of concepts developed in other ways. This is a kind of abstraction or extraction of the essential parts of a given concept. The symbols, as written in mathematical form, now represent the unique and essential aspect of the original insight. However, it must not be forgotten that most scientific insights do not come from mathematics itself, although this does happen as well, mostly such insights e.g. the idea of natural selection, are intuitive insights rather than related to mathematics as such. However, this does not mean to say that the mathematical way of thinking doesn't influence such insights.

On the relationship between mathematics and science-Part 2:

The 21st-century counterparts of these great figures - those seeking to mesh general relativity and quantum mechanics in a unified theory - are not so lucky. A unified theory is key unfinished business for science today.

The most favoured theory posits that the particles that make up atoms are all made up of tiny loops, or strings, that vibrate in a space with 10 or 11 dimensions. This string theory involves intensely complex mathematics that certainly cannot be found on the shelf, and the challenges it poses have been a stimulus for mathematics. Ed Witten, the acknowledged intellectual leader of string theory, ranks as a world-class mathematician, and several other leading mathematicians have been attracted by the challenge.

String theory is not the only approach to a unified theory, but it is by far the most intensively studied one. This endeavour is surely good for mathematics, but there is controversy about how good it is for physics. Arguments rage over whether string theory is right, whether it will ever engage with experiment, and even whether it is physics at all. There have even been commercially successful books rubbishing the idea.

To me, criticisms of string theory as an intellectual enterprise seem to be in poor taste. It is presumptuous to second-guess the judgement of people of acknowledged brilliance who choose to devote their research career to it. However, we should be concerned about the undue concentration of talent in one speculative field.

Finding a unified theory would be the completion of a programme that started with Newton. String theory, if correct, would also vindicate the vision of Einstein and the late American physicist John Wheeler that the world is essentially a geometrical structure.

--------------------------------------------------

Part1 and 2, New Scientist issue no. 2695, by Martin Rees.

Profile

Martin Rees is professor of cosmology and astrophysics and master of Trinity College at the University of Cambridge. He was appointed Astronomer Royal in 1995 and was President of the Royal Society between 2005 and 2010. The above is based on contributions to a discussion by a panel that included mathematicians Michael Atiyah and Alain Connes about the relationship between mathematics and science.

Note: I have also included this contribution in another related thread:

What is the relationship between Mathematics,Science and Nature?

https://www.researchgate.net/post/What_is_the_relationship_between_Mathematics_Science_and_Nature

Issam,

"String theory is not the only approach to a unified theory, but it is by far the most intensively studied one. "

It is not true anymore. The string program is rapidly slowing down. The hiring of string theorists in physics departments went down drastically in the last five years. The amont of publications of major string theorists went down drastically. At the same time quantum gravity programs such as Rovelli loop gravity are getting more attention.

Frank,

"Nowadays mathematics and science are strictly separate areas of study, "

It is true as far as acedemia is structured but it is not a good thing for many reasons. It is not at all a coincidence if the mathematics of quantum theory (Hilbert space) and of General Relativity which see gravity from a Rieman's geometry perspective. In the second half of the nineteen century, all the top mathematician involved in the revolution of the non-Euclidian geometry (Gauss,Poincarre, Hilber, Rieman) were also the to theoretical physicists of the time and they all knew that these new geometries would revolutionize physics. Already in 1901, Poincarre and Lorentz had invented 90% of special relativity. For mathematics to be relevant to physics it is necessary that the mathematician is also interested in solving the most pressing physical problems of his/her time.

Louis,

Thank you for the update.

Mikko,

If I understand the question correctly the answer is that Math hinders the development of pure science. We tend to look to math and its rigor to solve the problems. This stops true thought. Math gives us the answer or we think that math gives us the answer. The thought process is only that we need to use the correct formula and the answer will appear. Nothing could be further from the truth. In true science even as in Albert Einsteins day one used their mind to work through the problem and on to what it would look like if you were to solve the problem in the manor that you proposed and then you looked to formalize that with equations after you were convinced that the solution was correct. Math is the language of the explanation not the solution to the problem. This does not mean that math will not point to other things that we have not thought of yet because it will, but it in and of itself is not the solution.

This is the reason that we have not worked out the problems with gravity and the quantum mechanics not being compatible. It is the reason that we have a model of the universe that is expanding at an expanding rate and there must have been a big bang to start it that can not be explained by any mechanics that we know of today. If the universe started at the big bang then what force made it come apart? Not one can explain how something that deep inside the event horizon of a super massive black hole could have made that matter move outside that event horizon at such a speed that it was ten to the ten to the ten time the speed of light and then for some unknown reason slowed down and then after a time started speeding back up again to get where we are today.

So what happens is we use math and we forget reality as long as it works in the equation. The problem is hopeless. Math is one of the best tools and the worst problems in the tool box of science.

@ Arno I dont see inclusion in either direction. Lots of Science is not math, eg archeology does not seem very mathematizable to me. And pure math is more like literature sometimes..Experimental math is not always science but conjectures and guesses...

@Arno what about classification in biology? What is mathematical about the pedigree of dinosaurs? Still classification of species and the careful naming of various families is perfectly rigorous...

I am not an expert on archeology, but just simple common things suggest that mathematics is involved in archelogy. When an archaeologist finds a site to dig they have to grid it out and label/number the grids. When something is found it is located on the grid and given a number. The location of an object is as important as what the object is. These are all forms of math, so when you are doing diagrams, charts, mapping, and graphs in math you are getting skills that you will use in archaeology.

In fact mathematics in archeology is well known and simple google search will give you lots of references, even books on the subject. One such book is :

Mathematics in Archeology by Clive Orton, published by Cambridge University Press 1982, ISBN 0 521 28922 X.

It seems the discussion strayed too far from the Mikko original question (the way I understood it) and my response . My comments were not aimed at deep philosophical underpinnings or historical perspective of math (Chris) , but something more specific and mundane - the role of "formal" (abstract, Bourbaki-style) math in modern science. My answer was "negative", not just "zero" (= "useless"). Such claim needs further clarification.

Is "math. language" useful ("good" or "bad") for science?

The very question sounds pointless to me (is moon light "good" or "bad" ?). No language by itself is "good" or "bad", it all depends on HOW one uses it.

Of course, math could be useful, provided it's used properly. And for sciences (real ones) it means first and foremost its informal use, the less formal the better.

I challenge any scientist in this audience to bring me a single example of a scientific work not even a breakthrough, where "uniqueness/existence theorem" played role (something professional mathematicians would often spend their whole life).

And please, no general declarations about relativity and quantum theory. I know enough of those to assess the importance of "uniqueness/existence" in modern physics, chemistry or biology.

At best one'd say math abstractions are useless for the real science.

Yet I went further to claim they have "negative" input. The latter of course is limited, as most scientists use math informally, and their students are trained properly by the scientists themselves (excepting a few intro math classes). As for pure math training, it's dominated now by abstract and dogmatic style of XX century, which does little good to a student or researcher who wants to apply it to real world problems. That what I meant by "negative".

David: I already referred to the Goldstone theorem in the context of the Higgs Boson. There is also the Penrose-Hawking theorem on the gravitational singularity which guarantees the existence of the Big Bang. There is also Gödel's incompleteness theorem which Turing recast in his treatment of computability, which Church later showed to be mathematically equivalent. Of course, this gave rise to the Turing Machine - quite influential, one might say ... Thereare lots of other examples too I am sure - these just immediately sping to mind.

Arno: Common sense is a slippery concept. GRT (the general theory of relativity) looks decidedly weird, I agree. Nevertheless, it now counts widely as common sense since everyone uses SatNav! Relativity mixes up energy mass and space so that we can't any more talk simplistically about them. So? Does meaning drain away?

I think that Roger Penrose is right here, and your comment "physics is stuck" ought to be read in a Penrosean way. In his "The Road to Reality" he makes what is to my mind a very powerful argument that our current formulation of quantum mechanics is incomplete, and the account we have of the collapse of the wavefunction is formally unsatisfactory. There remains the problem that QM and GR are not integrated, and no-one knows how to do it. The only place they join in current theory is in the Bekenstein-Hawking formula for the entropy of a black hole.

I think, on reflection, that David Gurarie's comment that current pure maths training is "dominated now by [an] abstract and dogmatic style ... which does little good to a student or researcher who wants to apply it to real world problems" if true is valid. Sorry to have been so negative, David!

@David, I am very surprized by a mathematician making a frontal assault on abstraction. There has been a lot of critics on modern math. I was educated (say at college level) in the late seventies when it was all the rage. Being french the Bourbaki influence was of course very strong. Let me emphasize this:

If I had not seen abstract vector spaces at age 18, I do not think I would have become what I am today, ie an active coding theorist, using vector spaces over finite fields everyday. Born 20 years earlier I would have become a philosopher.

@Arno: u have a strong bias towards continuous math, but even applied maths like David use linear algebra everyday...

Set theory is borderline with logic

For instance, logic never won a Fields medal, hardly figures in Bourbaki set of books, etc...

The physical world is a hierarchical organization. It exists at different levels of organization and the laws existing at a level are not all reducible to the lower level. Trying to explain the behavior of boats in water with the laws of the particles which atoms are made off would not be very useful. Mathematics too has many levels build onto each other. Although all mathematics can be build from logic via set theory, it does not mean that all of math can be reduced to logic and set theory. Geometers do not necessary need to think in terms of these lower level mathematical concepts.

Related to the question, here's an interesting short read, titled "Less proof, more truth": http://www.cs.auckland.ac.nz/~chaitin/byers.html

Arno,

I was a bit at loss on how to answer your question. My initial intuition was that most of creative mathematics does not need to concern itself with foundational work in logics and set theory. The text "Less proof, More truth" goes straight at the spirit of what I was trying to say. Thanks Mikko.

Dear Arno,

“So what of our own, unheroic, age? Would Euler, Cantor and Ramanujan be welcome now? Definitely not. …

What went wrong? Well, it started around the end of the 19th century with David Hilbert's vision of complete formalism, of proofs so thorough a computer could check them. It was a vision widely propagated by the French Bourbaki school of mathematics, which, strangely enough, preferred a rigid, Prussian, vision of maths rather than their own more sensual tradition.”

Twentieth-century mathematics decide to eschew words, ideas, diagrams, examples, explanations and applications in favour of formulae. This is a lawyer's vision of maths, where the main goal is the nit-picking avoidance of mistakes."

I think that the author is giving to much credit to Hilbert because he focused too much on mathematics . Hilbert is more the symptom of the disease than its cause.

Mathematics is not alone, 20th century philosophy, science is mostly nit-picking in the sense that it is mostly a constellation of micro-contributions (papers) filling micro-gap of knowledge among papers. Every philosophers and scientists has to be productive in the short terms. Micro-gap filling is important but it cannot provide guidance and leadership outside current paradigm. New paradigm are kind of long bridge (large gap) across far apart area of knowledge.

“To create a new field of mathematics, you have to feel comfortable with paradox, with creative tension, with sloppy and dangerous new ideas, and you have to want to rock the boat, not conform slavishly.”

“It is time to free mathematical creativity from this prison. We need a radically new mathematics for our postmodern era, a mathematics of complexity, computation and information, a mathematics that applies to complex biological systems.”

“We have computers now, so we don't need to have people imitating machines. The 21st century is beginning: time to throw off our chains, and unleash the power of our imagination and creativity. “

Why do you think that it is not the rule but the exception? Is it that it is more difficult to be creative today than it used to be when mathematics and sciences and philosophy where less complex? Or is it because most scientist/mathematicians/philosophers do not feel in their guts the necessity to invent something very new?

In response to Chris, I never claimed that logic or abstract reasoning should never be applied in science. Both could be useful of course, when done appropriately (within limits). Most scientists are perfectly capable of doing it without undue "overdose" of abstract formalism. When grand sounding ("mathematical") names are given to scientific principles (Goldstone, Penrose-Hawking theorem, Gödel's incompleteness) in practice it could mean different things for a physicist ot a mathematician. With all due respect I doubt "Goldstone theorem" would take more than half an hour in a typical physics class to explain its essence (meaning and contents). In contracts, an abstract mathematician could spend a large chunk of semester on its formal proof with little else to gain. Indeed, any physical subject (be it particles, GRT, cosmology) has a great deal more important material to cover (besides Goldston , Penrose-Hawking) that indulging in mathematical proofs and derivations becomes prohibitive. As for "Gödel's incompleteness" (a great delight of logicians and algebraists), I doubt it gets mentioned in any computer science class.

Overall, excess formalism detracts rather than helps scientists.

In response to Patrick, my personal experience was different from yours. Having been trained like you, on Bourbaki, I found it useless later in my career when switching to applied fields. Not only I had to learn "useful" math and science on my own, I also had to retrain myself from the abstract/forma" way of thinking.

Arno,

Godel's incompleteness must apply to all fields if it applies to any.

Last Friday, Polly Toynbee said in the Guardian (she is an excellent columnist in an excellent UK newspaper) that "the here and now is all there is". As I interpret it, Gödel's incompleteness theorems are a proof that she is talking uncharacteristic nonsense. To say that we can specify "all there is" is logically incoherent. Axiomatic knowledge (and that includes physics!) is necessarily incomplete. There always exist true things that can be properly described in a system which have no proof in that system. The boundaries of the system must be stretched. This is necessarily true of all sufficiently complex systems, and arithmetic is sufficiently complex.

The point of formal reasoning is that it underpins all other reasoning. Probably there is not one Christian in a million who has read and understood Augustine's treatise on the Trinity. Nevertheless, Augustine's reasoning underpins that Christian doctrine which has been accepted for 1600 years, and which is still common currency. By the way, this last point addresses Mikko's original question (can rigor be useful or even implemented in other fields?)!

I accept David Gurarie's points. Doubtless a methodological overemphasis on rigour hinders progress. But every overemphasis will hinder progress! However, I maintain that the existence of rigour underpins any argument and promotes progress in that what is known and unknown has become better understood.

If rigour is properly applied it restricts the axioms one can properly use. The mathematicians do us a service by clarifying our language.

Mikkos's question asks: "How does mathematical rigor influence Science?"

However, the opposite question, how does science influence mathematical rigor?, is a legitimate one. Here is an example that progress in physics was crucial to rigorous mathematical solutions of questions in mathematics not originated in physics.

The understanding of quantum anomalies came directly from a physical observation (the high rate of neutral pion decay to two photons). Clarifying how this happens through Feynman diagrams, even though a naive argument says it is forbidden led to complete understanding of all anomalous terms in terms of topology. This in turn led to the development of Chern-Simons theory, and the connection with Knot polynomials, discovered by Witten, and earning him a Fields medal.

Arno: There is thread on RG that discusses this topic

"If axiomatic physics is possible, what would be the axioms? "

https://www.researchgate.net/post/If_axiomatic_physics_is_possible_what_would_be_the_axioms

Probably mathematical rigor is not as significant in Physics or applied sciences as is an insight and common sense logic. In some situations where fundamental laws are involved, mathematical rigor can be important. Otherwise, I find many physicists struggling to interpret and understand the jargon a solution of mathematical equations throw without appreciation the physical situations.

A mathematician belives nothing until it is proven

A physicist believes everything until it is proven wrong

A chemist doesn't care

biologist doesn't understand the question.

--------------------------

Biologists think they are biochemists,

Biochemists think they are Physical Chemists,

Physical Chemists think they are Physicists,

Physicists think they are Gods,

And God thinks he is a Mathematician.

To Arno question "does Godel's theorem applies to physics?"

If you ask any practicing physicist (or other scientist) the answer would be definite NO. Even within abstract math its use is limited to logic and algebra. In fact, I daresay philosophers are more interested in Godel than mathematicians.

To Issam:

Feynman developed his diagram techniques as a tool for "quantum field" calculations. I doubt he cared much about Chern-Simons, knots, Witten strings, or other abstractions that followed. Of all subjects in modern theoretical physics "string theory" is the most mathematical (not surprising Witten got his Fields). But (not my opinion only) it's the least "physical". A hallmark of good science is to make verifiable predictions, and to reach across other areas/disciplines in terms of methods and ideas. After 40 years of heavy "math development" string theory has little to show on both counts.

David

Quite right. I was trying to show that not only mathematical roigor has an influence on science(mainly physics) but also sometimes progress in physics was crucial to rigorous mathematical solutions of questions in mathematics not originated in physics. I was not at all showing the relevance of mathematical abstractions to the real(physical) world.

There is a thread on RG talking about similar issue:

"Are there any mathematics for which there is absolutely no application in physics?"

https://www.researchgate.net/post/Are_there_any_mathematics_for_which_there_is_absolutely_no_application_in_physics

It would be interesting to hear your views on this subject.

I should mention that the above question has a critical bearing on the development of biology: a very poor organizational state of biology, including a superficial view of evolution, is a direct consequence of the lack of formal, or structural, training of biologists.

In general, the emergence and evolution of life is a much more abstract subject than that of the present day physics. One cannot approach it without even more 'abstract' and 'rigorous' training than that of physicists.

David G,

There are in fact many mathematicians and scientists who appreciate on the record that the 'bigger picture' takeaway from Gödel is that closed systems have a fundamental property with has direct consequences in physics.

As for String theory, it still does have a number of things to show for itself.

Beyond immensely strengthening the mathematical abilities of a generation of physicists - no mean feat - it has led to insights such as solving the black hole entropy puzzle which resisted solving by any other means. It most certainly won't be the end-all theory of physics, but the effort extended over the last years on ST has already paid dividends, which will likely be put to great use elsewhere.

Juan-Esteban Palomar Tarancon - your distinction fascinated me! I like it. Hope biologists are not offended.But why God rated mathematicians above physicists? (I am a physicist).

@Jindal,

"But why God rated mathematicians above physicists?"

For quite obvious reasons: without a *useful* formal language we can't really do physics. (Of course, the majority of today's mathematicians are not of that kind. ;--) )

Math is the language, Physics is the solution, and the human mind solves the problems not math. I will give you that math can and has lead use to solutions that had not been thought of yet, but we would not have found the problems or solved them with math had we not thought through the process with our minds.

V K Jindal

"But why God rated mathematicians above physicists?" This is why:

“God used beautiful mathematics in creating the world.”

---Paul Dirac

“Mathematics is the language with which God has written the universe.”

― Galileo Galilei

I am aware that not all mathematicians believe in God, including Paul Dirac himself!!

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

These quotes are not mentioned here to illustrate God's existence-this is not the issue here- but to illustrate the mathematical nature of the universe.

All the above discussions are interesting and up-to-mark.