Good question.

The amazing thing about mathematics is that it is applicable in the empirical sciences. If , for example, we are talking about geometry, points are dimensionless and lines have zero breadth. So the points and lines familiar to scientists who study shapes and geometric configurations have approximate relation to mathematics.

Incredibly interesting question, the missing link is in the concept of 'representation': a portrait of someone made by a great baroque painter (see atteched the Van Dyck portrait of Cornelis van der Geest one of my favourite portraits..)

work or, to be more precise, gives us the possibility to appreciate a TRUE and REAL piece of information just because IT IS NOT MADE of the same matter of the reality it is supposed to represent, but because, on the contray, is a partal BUT HONEST and more important DELIBERATE point of view. Thus we know after 400 hundrede years a lot of things about Cornelis thanks to van Dyck that had a deliberate idea of his subject that IS NOT HIGHLY FORMALIZED but on the contrary, really passionate and material. The correct use of mathematics in science (in the cases in which it works) is like a baroque portrait: it allows to use a deliberate and personal point of view on the reality, in those cases it works, when is too formal (and uses phrases like 'it must be true because the mathematics is correct') it is not good art, it is like the very cold 'classicist' portraits of the end of XVIII century....rubbish, they do not work at all...

Alessandro,

Excellent post! I agree. It is representation that defines the relation between mathematics and the empirical sciences.

Consider this: no matter how minute detailed the description T of a situation S is, how adequate the language in which it is written happens to be (provided it is a language, mathematical or otherwise), there will always be a materially different, but formally identical situation S' (an isomorphic copy od S) of which T is also true. Hence, no linguistic description can "capture" eveything so as to singularize a situation. T can only "grasp" the abstract form of S, which S' also possesses. Only the formal surface of the experienced world finds its way into science. How much of this is only a net we throw over reality for being the way we are, which only filters reality according to the patterns of perception and conceptualization that we are capable of?

No singular event can be captured, we can only work by categories, this was formally established in XIII century by the Franciscan philosopher Duns Scoto, you are totally right, but this is not a particular problem to me, on the contrary the idea of capturing everything is a form of unbearable arrogance, Jorge Luis Borges explained this point very well in the below 'short-novel' entitled 'El Rigor de la Ciencia':

(On Exactitude in Science):

. . . In that Empire, the Art of Cartography attained such Perfection that the map of a single Province occupied the entirety of a City, and the map of the Empire, the entirety of a Province. In time, those Unconscionable Maps no longer satisfied, and the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it. The following Generations, who were not so fond of the Study of Cartography as their Forebears had been, saw that that vast map was Useless, and not without some Pitilessness was it, that they delivered it up to the Inclemencies of Sun and Winters. In the Deserts of the West, still today, there are Tattered Ruins of that Map, inhabited by Animals and Beggars; in all the Land there is no other Relic of the Disciplines of Geography.

Grazie per i tuoi commenti, Alessandro! Mi piacerebbe continuare in italiano, la lingua dei miei bisnonni, ma come questo è un sito internazionale, avanti con l'inglese.

But there is also Funes, el memorioso, to whom nothing ever scapes.

I don't see it as a problem either, it is just how things are. My point is that this is a start in explaining how mathemaitcs can be of relevance for science. Since empirical science has to do only with the formal shell of experienced reality, and mathematics is nothing but the science of formal structures, the formal structuring of experience can somehow find its way into mathematics. My problem is how.

Caro Jairo, complimenti per l'ottimo italiano !

You say that your problem is 'how to use formal structures to describe empirical sciences'' my answer is to use less formal as possible (or if you prefer informal) structures. A good example is in Principal Component Analysis and Recurrence Quantification Analysis (see attached file). in PCA you work as a 'capocomico' of the old 'Commedia dell'Arte' : you have a set of fixed 'characters' that you can apply to describe virtually any situation. In the Commedia you had: the young actor (typically the 'lover'), the young actress (the sweet and beautiful, often poor and delicate, girl), the 'bad' old guy (in some instances a sex-maniac and rich guy that tries to seduce the young girl), the 'good' and smart helper (a friend of the young actor helping the young guys to go into their 'love dream' and defeating and putting into ridiculous the bad-old-guy).

In PCA you have the 'order parameters' hierarchically organized in terms of relative importance (variance explained = amplitude of the attraction basin for the maniac of physics..) corresponding to the eigenvectors of the correlation matrix among original descriptors of the system. The specific role (script) played by each eigenvector (principal component) can be described by the loadings (correlation coeffcients with original variables), that's all, you do not impose any structure to your data but let the play to emerge from the empirical correlation existing among variables. Clearly here the 'Art' is in finding a good data set and good intial descriptors. It is 30 years I work with the same scheme going from protein science (see 'protein as sponges' in the forefront of my RG page) to animal behaviour, medicinal chemistry, botany....and it worked !

....you could be interested into the attached paper too (a very good primer on PCA)

Thank you for the reading material, which I'll read as soon as time allows. I loved the analogy with Commedia dell'Arte! But let me reestate "my problem". Not so much "how" in the sense of modus operandi, but in the sense of "how come?" I take for granted that math is applicable in science in many different ways and according to many different strategies. What does it tell us about science, nature and math itself? Sarò in Sicilia in febbraio per 12 giorni per la prima volta, che c'è da non perdere assolutamente? Buon Anno!

Palermo, Etna, Catania, Siracusa, Ragusa Ibla, Modica, Cefalù, Segesta, Taormina, Caltagirone, Piazza Armerina, Cava d'Ispica, Scopello e Riserva dello Zingaro, Saline di Trapani, Necropoli di Pantalica, Selinunte

''How come' I do not know, probably ha to do with the fact Universe is rational just like our minds and mathematics ..

Thanks for the suggestions.

That's it! The answer I cannot accept, that the Universe is intrinsically rational. There must be a non-anthopocentric answer!

The idea that the Universe is rational is antiscientific; it is on a par with "final causes" in physics or "vis viva" in biology. If we accept it, we must also accept either the idea of an "intelligent maker" of the Universe or Spinoza's "Deus sive Natura". This is the way Eugene Wigner, for instance, goes. But it is antiscientific. No scientific explanation of the effectiveness of mathematics in science can presuppose that Nature is tailormade for mathematical treatment. This is the philosophical part of the problem. But there is also a methodological aspect to it: what are the formal relations between abstract structures that guarantee that structure M ( M for "mathematics") can be effective in the investigation of structure N (N for "nature")? One possibility: M and N are isomorphic. But it is not the only one. Which others?

Why do you make such a chain of consequences ? I believe in God and he is separated from the Nature that is our sister not our mother, Spinoza was only a very old hippie..and the tougths of God are not are thoughts.. in any case even mathematics is empirical in its birth : I think first came someone in the paleolithic that had two stones in his hands and decided they were equally weighted and after so much time someone else developed the sign = to express this sensation...thus mathematics is material just as the stones, but we forgot this birth and we prefer to think there is something 'rational' implying rational is something else than empirical when at the basis there was the 'ratio' of two very material stones...

This is the reason why I prefer principal component analysis over many other things: it still smells of reality.....

A last word something that is not scientific is not necessarily antiscientific, by bthe way I'm not chinese but I am not anti-chinese ! Science is a part of our humanity not the whole....

Mathematics has an explanatory as well as representation role in empirical sciences. Here is an example:

Why is it that the spacing and intensities of the bows in rainbows are the same. The explanation for this phenomenon stems from the sizes and shapes of raindrops that will vary with each rain shower. To explain this universal pattern in rainbows, we consider the wavelengths of light in the limit as the wavelength goes to zero. With zero wavelengths, we enter the domain of geometrical optics and the stability properties of ray structures explain the universal rainbow pattern.

This example comes from a paper by R.W. Batterman, University of Western Ontario, 2008 (see attached pdf file).

Some people claim that mathematics has also a heuristic role in science, that the juggling of often meaningless mathematical symbols can sometimes lead to scientific discoveries. Examples often cited are Maxwell's "discovery" of displacement current and Dirac's "discovery" of anti-matter (scare quotes are used for a reason). I cannot agree and can tell different stories in both cases.

Dear Jairo,

Good question! It makes me think about 2 things:

1, mathematicians are like painters, mathematicians’ works are similar to pictures by painters; how much they "model" reality？

2，we operate in our mathematical way------- mathematical culture.

Regards

There is much more about mathematics being close to art than people in general realize. Like novels, mathematical theories often refer to realms of fantasy that, nonetheless, help us to understand those of reality.   

Dear Jairo,

We come to the bottom: What human mathematics is! It is one of human’s sciences.

We human exist with our own time and space while Universe and Nature exists with its own time and space; this is one important fact we have to have in our mind all the time. Existing along with human, human science is human’s cognitive product to Universe and Nature------things in our science should be with the property of “human time and space” and “Universe and Nature time and space” so we know how they are from and how they are being in our science.

Thus, we have "reality" and "model", "infinite" and "finite", "homogeneous" and "fractal", "consistent" and "inconsistent", ….

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