Dear Juan,

I entirely agree with you that social sciences are tremendously complicated compared to maths and physics, with biology somewhere in the middle. In my opinion it is because man is simultaneously the observer and the observed subject of these sciences, while for the others the observed matter is outside, for maths and physics, or partially outside for biology.

As Granger (1994) said: In the case of social sciences, the transmutation into a scientific object of the complex and changing life experience that consists of the human fact remains problematic, even in its aspects commonly recognized as public. This is not because a prior and general explicit definition of the object of a science is a prerequisite for its development. Quite to the contrary, physics and biology are, if anything, progressive explanations of their object. But it is no less true that, in these conditions, every objectivated aspect of sensitive experience, every phenomenon, is identified by means of criteria accessible to all those who possess a specific physical and intellectual tool set. These criteria, however refined, mediated, and abstract they may seem, nevertheless ‘salvage’ phenomena, in the sense that they always allow, within the limits of a recognized tolerance, the establishment of a univocal correspondence of the phenomena perceived, and consequently experienced by a private conscience, with their publicly intelligible schematization. However I think that this transmutation had been done when for example Durkheim (1895) used the concomitant-variation method in order to explain some very precise social facts, like suicide. He said: The reason is that, for [the method] to be demonstrative, there is no need to strictly exclude all of the variations that differ from the ones being compared. The simple parallelism of the values taken by the two phenomena, provided that it has been established in a sufficient number of sufficiently varied cases, is the proof of a relationship between the two. The method owes this privilege to the fact that it arrives at the causal relationship not from the outside, as with the previous methods, but from the inside. By examining separately each of the various social, political, religious and other characteristics capable of influencing a phenomenon, it becomes possible to dissect their effect on it. However this approach is subject to the ecological fallacy and needs to be completed.

References

Durkheim, E. (1895). Les règles de la méthode sociologique. Paris : Alcan.

Granger, G.-G. (1994). Formes, opérations, objets. Paris : Librairie Philosophique Vrin.

Dear Daniel,

First, there is what is called social sciences which is different from natural sciences. Natural sciences obey strict laws of nature while social sciences follow human made laws, conventions, cultures and traditions which in most  cases are backwards and liable to change.

In regard to mathematics and other natural sciences, it is always the case that other natural sciences observe natural phenomena through intuition, predict its behavior, make an empirical tests and validate its truthfulness through mathematical rigors - the very reason we say mathematics is the language of nature and the existence of those mathematical hyphenated natural sciences mentioned by dear George.   

This is a good question.

@George Stoica:...the old branches found (and are still finding) bridges to connect their common interest.    I agree.

Even so, there are fundamental differences between mathematics, physical, biological and social science.   For example:

Pure mathematics considers abstract structures independent of the physical world.   Examples are algebraic structures (groups, rings, fields), topological structures (open sets, closure of a nonempty set, dense spaces).

Physical sciences consider physical structures and systems such as light, planetary bodies, molecules.

Biological sciences considers animal and plant structures and systems such circulatory systems, vision, tissue cells, plant cells.

Social sciences consider the behaviour and interactions of societies.

Dear George,

I clearly understand your point of view: mathematicians and other scientists are always trying to find a common language in order to solve specific problems. It seems to me that this language is mainly mathematical, and more precisely for the social sciences it is the language of probability which offers this bridge. Interestingly this concept appeared with Pascal and Fermat exchanges in 1654 and was named by Pascal The geometry of chance, while the concept of social science appeared in 1662 with Graunt’s book on Natural and political observations.

It was only at the beginning of the 20th century that Hilbert in his famous list of the 23 problems to be addressed said: The investigations on the foundations of geometry suggest the problem: to treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part: in the first rank are the theory of probabilities and mechanics. In 1933 Kolmogorov put this axiomatization in practice for objective probability, in 1937 this was made for subjective probability by de Finetti and in 1939 this was made by Jeffreys for logical probability.

However for social science, even if there were some attempts of axiomatization, we cannot say that these social scientists succeeded to do it.

I will try to answer more precisely to the other interesting comments a little later.

References

Finetti de, B. (1937). La prévision : ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincaré, 7, Paris, pp. 1-68.

Hilbert, D. (1902). Mathematical problems. Bulletin of the American Mathematical Society, 8 (2), pp. 437-439.

Jeffreys, H. (1939). Theory of probability. New York: Clarendon Press.

Kolmogorov, A. (1933). Grundbegriffe der wahrscheinlichkeitsrenung. In Ergebisne der Mathematik, vol.2, Berlin: Springer (English translation, 1950, Morrison N. Foundations of the theory of probability, New York: Chelsea).

Dear James,

I entirely agree with you that there are fundamental differences between mathematics, physical, biological and social sciences and you had clearly shown how they differ. Even I think that the different social sciences like sociology, demography, economics, and human geography show important differences in their object. The same is also true for the different other great branches of science like mathematics, physics or biology.

However we can see that the foundations of the major part of these sciences are linked to the existence of axioms, different for each of them, but which have the same signification: they are abstract concepts without which the observed properties of a system studied by a science would be inconceivable or impossible, as Franck (2002) said. Such a task of discovering the axioms of a given science was already present in Euclid’s time and is always present nowadays. I think that this task which had been accomplished for many other mathematical, physical, biological sciences has not yet been done for the major part of social sciences.

Reference

Franck, R., Ed. (2002). The explanatory power of models. Bridging the gap between empirical and theoretical research in the social sciences. Boston / Dordrecht / London: Kluwer Academic Publishers.

Dear Dejenie,

I entirely agree with you that natural sciences are different from social sciences for the reasons already given in my answer to James.

However I may question your definitions of natural and social sciences. First I am not so sure that natural sciences obey strict laws of nature.

For example by viewing the universe as probabilistic instead of certain, quantum physics leads to a weird world very different from what we see. As Merali (2015) said: a wave of experiments is probing the root of quantum weirdness. The wave function permits to calculate the probability of observing for example an electron at any particular location, but the mathematics shed no light of whatever a wave function truly is.

Conversely a very recent research result by Markram et al. (2015) permits, using reverse engineering, a digital reconstruction of the microcircuitry of the cortex of juvenile rat. Even if this is only a first step in exploring animal and human behaviour, I think that it shows clearly that reverse engineering, by succeeding in reproducing the properties of a system of 30000 neurones connected by around 40 billions points of contacts, makes possible the discovery of the system’s true functional structure.

The discovery of a system structure was termed induction by Francis Bacon in 1620, which is deeper than the conception of induction as generalisation from particular instances. Induction then consists of finding the principles or axioms from the study of their consequences. It is in this way that I think that the field of social sciences may be more precisely defined.

References

Bacon, F. 1620. Novum Organum. London: J. Bill.

Markram, H. 2015. Reconstruction and simulation of neocortical microcircuitry, Cell, 163, pp. 456-492.

Merali, Z. 2015. What is really real? Nature, 521, pp. 278-280.

Dear Marković,

May be did you read the recent paper on Population science, philosophy and art, I presented in my featured publications on ResearchGate? You will find in this paper my point of view on the links existing not only between philosophy and science but also with art. I think that you will be interested to read the paper published by Lucatelli (2007) on religion, art and science.

However, even if mathematics is born simultaneously with philosophy particularly with the work of Euclid, I don’t think like you that it is a special science. For me, Euclid’s common notions are axioms, which have been extended to set theory axioms or to probability axoms, as in every natural science.

Reference

Lucatelli, F. 2007. Why we disagree: modal conflict among religion, art and science, Journal for Interdisciplinary Research on Religion and Science, 1, pp. 259-274.

Dear George, James, Dejenie, Marković and all other researchers interested in this question,

I would like first to thank you for this first round of answers to this question, which permitted us to go more deeply into its roots. As you can see it needs more thoughts about the way to follow in order to explain a mathematical, physical, biological or social phenomenon. Do we need a theoretical explanation, in order to see what structure is necessary for the properties of the phenomenon, or do we need a more empirical explanation to see what mechanism generates it? Or are the two explanations necessary in order to understand this phenomenon?

I hope also that other researchers will try to give answers or comments to these questions: I will be very interested in all their ideas and will be happy to see their developments.

More or less obvious answer to the original question is 'yes', there is a fundamental difference between mathematics and physics on one side and the remaining disciplines on the other.  The difference is life; living matter obeys physical laws but it is also able to behave in a way not described by physics.  For example, I may suddenly decide not to eat today or I can change my mind and go to sleep rather than watch TV as I was planning yesterday.  I'm not sure whether such a distinction fully applies to plants or to bacteria.  On the larger scale (sociophysics) physical laws seem to apply to living things.  One can say: it's a triumph of statistics, which is a part of mathematics.  Anyway, we have serious troubles with exhaustive (physical) description of a single animal.

Some will say that the grand unification is religion.  But even then: which one?  And why this one not that?

I think the good answer for the question posted will be possible only after we know "what is life?" (Erwin Schroedinger).

@Marek Wojciech Gutowski: ... The difference is life; living matter obeys physical laws but it is also able to behave in a way not described by physics.

Yes, great insight in that observation: life (living systems) provides a fundamental difference between living matter in physical systems and the sciences (mathematical, physical, biological and social).    It would be good to add here that the sciences observe life, whereas life itself is shrouded in mystery, persisting and evolving in a physical as opposed to a theoretical way.

Dear Marek,

I entirely agree that the good answer to the question will be easier to give if we know “what is life?”. However Schrödinger did not think that physics and living matter were so different. He wrote in Chapter VII of his book: We must therefore not be discouraged by the difficulty of interpreting life by the ordinary laws of physics. For that is just what is to be expected from the knowledge we have gained of the structure of living matter. We must be prepared to find a new type of physical law prevailing in it. Or are we to term it a non-physical, not to say a super-physical, law? I think that axiomatization may be applicable as well to physical sciences as to biological and social science. Evidently this is not an easy task, but like Schrödinger I think that it may be possible.

Dear Daniel,

I, too, hope that the connection between physical and biological sciences is going to be more and more tight in near future.  However, for this to happen we need some new  paradigm; completely new insight; call it axiomatization if you like.  It may be quite near, since the same Erwin said:   The task is ... not so much to see what no one has yet seen, but to think what nobody has yet thought, about that which everybody sees  (Dublin Lectures, 1943).  Perhaps a great mind able to perform this formidable task was already born?  Who knows ...

As for today, we are aware that DNA molecule, even seriously "damaged" (one half of this 2 meters long molecule) still remains a carrier of life, not being a living object itself.  But we do not know how to distinguish between life and lack of it.  Some worms, after being halved, remain alive - but this is not the case for, say, pigs.  We need to understand (solve?) this mystery, first of all.  Schrödinger was fully aware of this.

Dear James,

Yes, life is able to behave in a way not described by physics, while obeying physical laws. More generally, don’t you think that all sciences (mathematical, physical, biological and social) retain only a little number of aspects of the studied phenomena which can be characterised by a small number of parameters? They consider the effect of other parameters as negligible in first instance and elaborate experiences trying to suppress entirely these effects. For example Newton’s mechanics considers only abstract points in space, distances and forces in order to study the motion of bodies. In this case, it was not able to explain the precession of the perihelion of Mercury, which is finally perfectly explained with Einstein general relativity. For demography, it was not the study of births, deaths and migration flows, which has been envisaged since antiquity without ever producing a truly approach in population sciences, but the study of the probability of fertility, mortality, and migration, which permitted the foundation of population sciences by Graunt in1662. I think that a theory is a filter that retains from complex phenomena only that which must figure in the object of the research.

Reference

Graunt, J. 1662. Natural and political observations ..., London: Roycoft.

@Daniel Courgeau: ...theory is a filter that retains from complex phenomena only that which must figure in the object of the research.

Yes, I agree.   Put another way, the basic approach in the sciences is to abstract away from many of the details of physical phenomena, focusing on those quantifiable aspects of phenomena that provide a basis for an appropriate model.   Your examples are great!   Namely, Newton's mechanics with its interest in spatial objects, distances and forces and Demography with its interest in the probability of fertility, mortality and migration.

With the mathematics, the situation is a bit different.   By that, I mean that mathematics is inspired by physical phenomena such tori, bouncing falls, stretchable surfaces, closeness of molecules and the apartness of the planets.   From these sources of inspiration, mathematics focuses on timeless structures such as topological spaces and mappings that deform one shape into another shapes.   From mathematics, we have structures such as a manifold each of whose neighbourhoods to points map onto a subset of an n-dimensional Euclidean space.   I firmly believe that the perception of the closeness of bunches of physical objects inspired the introduction of the proximity of sets of points.

Strangely enough, these disparate points of view in science and mathematics come together in ancient philosophy in works of Plato (forms), Artistotle (essential being), Democritus (change), Parmenides (persistence).   This habit of the sciences in abstracting from physical phenomena also appears to have influenced many of the great discoveries in mathematics, starting with Archimedes (approximation) and Euclid (geometry).   See, for example,

https://www.researchgate.net/publication/27477716_Archimedes_revisited_A_faster_better_cheaper_method_of_accurately_measuring_the_volume_of_small_objects

and

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

and

https://www.researchgate.net/publication/225761026_Euclids_fifth_postulate

Article Archimedes revisited: A faster, better, cheaper method of ac...

Article Euclid’s fifth postulate

Dear James,

I entirely agree with you when you say that: Strangely enough, these disparate points of view in science and mathematics come together in ancient philosophy …, and your interesting examples lead me to ask for a full-text of the papers by Stephen Hughes and Renuka Ravindran.

I think that these ancient mathematicians and philosophers were more conversant with the origin of the axiomatic of geometry by Euclid, than we may be nowadays. I recently read a vey interesting book by Hintikka and Remes (1974) which provides a clearer view of Greek mathematicians’ method of discovery, which goes beyond a formally correct proof. Pappus, who had in the fourth century A.C. a thorough knowledge of the history this method of discovery, dating from the fourth century B.C., wrote: For in analysis we suppose that which is sought to be already done, and we inquire from what it results, and again what is the antecedent of the latter, until we on our backward way light upon something already known and being first in order. And we call such a method analysis, as being a solution backwards. In synthesis, on the other hand, w suppose that which was reached last in analysis to be already done, and arranging in their natural order as consequents the former antecedents and linking them one with another, we in end arrive a the construction of the thing sought. And this we call synthesis. They then clearly show that Newton was analysing an experimental situation in the same way as Pappus was establishing the interrelations of several parts of a geometric figure. Newton wrote in his Opticks (1704): As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition. This Analysis consists in making Experiments and Observations, and in drawing general Conclusions from them by Induction, and admitting of no Objections against the Conclusions, but such as are taken from Experiments, or other certain Truths. For Hypotheses are not to be regarded in experimental Philosophy. And although the arguing from Experiments and Observations by Induction be no Demonstration of general Conclusions; yet it is the best way of arguing which the Nature of Things admits of, and may be looked upon as so much the stronger, by how much the Induction is more general.

So that if the difference between mathematics and physics is in their object, the method of analysis and synthesis remains the same for these two sciences.

Reference

Hintikka, J., Remes U. 1974. The method of analysis, Dordrecht: Kluwer.

Newton, I. 1704. Opticks, London: Smith and Walford.

I would say that the social studies is  a tremendously complicated

topic compared to math and physics, with biology somewhere in the middle.

That is why the usuall perception is that more progress is made in the first two,

That does not have to stop anyone from choosing well defined problems in social science, and make important progress.

Hard areas in physics, such as particle physics, have been compared to trying to solve social problems.

@Daniel Courgeau:

Pappus: ...

For in analysis we suppose that which is sought to be already done, and we inquire from what it results, and again what is the antecedent of the latter, until we on our backward way light upon something already known and being first in order.

Newton: ...

This Analysis consists in making Experiments and Observations, and in drawing general Conclusions from them by Induction, and admitting of no Objections against the Conclusions, but such as are taken from Experiments, or other certain Truths.

The two view of analysis are subtly different in one way and similar in another way:

Difference in two views of analysis: supposing what we seek has been done (Pappus) and about the results of what has been done (Pappus) vs. experiment, observe and check our hypotheses, make assertions about the experimental results (Newton).   With Pappus, we check what has been done and with Newton we conduct experiments, which are the twin sources of our knowledge.   It looks like Pappus' view of analysis is historical, whereas Newton's view of analysis is more eclectic, probing, inspired by uncertaintly about any hypothesis we make.

Similarity in two views of analysis:  Both Pappus and Newton rely on sense data and evidence either from the past or from experiments as a source of knowledge and as a path to corroborating our theories.

 Im afraid to say that is some of the more speculative branches of physics, evidence as such has all but been forgotten, and math taken over. One can see this in supersymmetry, string theory, m theory, twistor theory, and in the almost hopeless area of trying to marry the general theory of relativity to the quantum theory.

Maybe we should have to start to revert this, since almost no progress is made since about 1930 or so in physics.

Dear Juan,

I entirely agree with you that social sciences are tremendously complicated compared to maths and physics, with biology somewhere in the middle. In my opinion it is because man is simultaneously the observer and the observed subject of these sciences, while for the others the observed matter is outside, for maths and physics, or partially outside for biology.

As Granger (1994) said: In the case of social sciences, the transmutation into a scientific object of the complex and changing life experience that consists of the human fact remains problematic, even in its aspects commonly recognized as public. This is not because a prior and general explicit definition of the object of a science is a prerequisite for its development. Quite to the contrary, physics and biology are, if anything, progressive explanations of their object. But it is no less true that, in these conditions, every objectivated aspect of sensitive experience, every phenomenon, is identified by means of criteria accessible to all those who possess a specific physical and intellectual tool set. These criteria, however refined, mediated, and abstract they may seem, nevertheless ‘salvage’ phenomena, in the sense that they always allow, within the limits of a recognized tolerance, the establishment of a univocal correspondence of the phenomena perceived, and consequently experienced by a private conscience, with their publicly intelligible schematization. However I think that this transmutation had been done when for example Durkheim (1895) used the concomitant-variation method in order to explain some very precise social facts, like suicide. He said: The reason is that, for [the method] to be demonstrative, there is no need to strictly exclude all of the variations that differ from the ones being compared. The simple parallelism of the values taken by the two phenomena, provided that it has been established in a sufficient number of sufficiently varied cases, is the proof of a relationship between the two. The method owes this privilege to the fact that it arrives at the causal relationship not from the outside, as with the previous methods, but from the inside. By examining separately each of the various social, political, religious and other characteristics capable of influencing a phenomenon, it becomes possible to dissect their effect on it. However this approach is subject to the ecological fallacy and needs to be completed.

References

Durkheim, E. (1895). Les règles de la méthode sociologique. Paris : Alcan.

Granger, G.-G. (1994). Formes, opérations, objets. Paris : Librairie Philosophique Vrin.

Dear James,

I agree with you that the answers of Pappus and Newton are only in one way similar and in another way subtly different.

They appear to be very similar when we read the sentences following those I already quoted in his Opticks: And if no Exception occur from Phænomena, the Conclusion may be pronounced generally. But if at any time afterwards any Exception shall occur from Experiments, it may then begin to be pronounced with such Exceptions as occur. By this way of Analysis we may proceed from Compounds to Ingredients, and from Motions to the Forces producing them; and in general, from Effects to their Causes, and from particular Causes to more general ones, till the Argument end in the most general. This is the Method of Analysis: And the Synthesis consists in assuming the Causes discover'd, and establish'd as Principles, and by them explaining the Phænomena proceeding from them, and proving the Explanations. Here Newton goes as far as to identify his own experimental method with the analytical method of geometers.

However they are different as you said that: With Pappus, we check what has been done and with Newton we conduct experiments, which are the two sources of our knowledge. I entirely agree with yyou on this point.

Dear Daniel,

What you have pointed out is important and tremendously interesting!   Many thanks for digging deeper into what Newton wrote in his Opticks.

I just now found a downloadable copy of Newton's Opticks:

http://www.relativitycalculator.com/pdfs/Opticks_by_Sir_Isaac_Newton.pdf

and a scan from the original manuscript:

http://brizhell.org/Newton_Speckles/Opticks%20-%20scan%20from%20original%20manuscript%20-%20I.%20Newton.pdf

A good example of what you have pointed out about Newton's scientific methods is in Part II, Prop. 1, theor. 1.   Newton's tests hypothesis by philosophers that light is variously modified by streaming sunlight through an oblong hole in a dark chamber and passing the light through a prism (see attached image for the geometry of Newton's experiment).

Dear James,

I have a copy of a first paper published in The Philosophical Transactions in 1672, which did not have the more general conclusions, and the full Treatise published by the Project Gutemberg, wich you can find in Google.  

The fundamental commonality admist the difference with respect to phenomenon of interest is the idea of geometry. In a sense the task of integrating this fields is the task of a geometer. To capture social, natural, mathematical and biological realities requires representational schemes, or "geometries".

https://origins.osu.edu/milestones/december-2012-god-particle-isaac-newton-s-birthday

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