You need to consider that population science also includes population biology and population ecology. They share a common foundation with demography (Lotka theory) but are much more developed in terms of axiomatics, especially in terms of biological and environmental determinants.

I think that your answer raises an important point. Yes, I entirely agree with you that population science shares a common foundation with population biology and population ecology. For population biology, for example, Illari and Williamson, in their 2010 paper published in Studies in History and Philosophy of Biological and Biomedical Sciences (see attachment) on Function and organization: Comparing the mechanisms of protein synthesis and natural selection, seem to augur well of the implementation of ‘Mechanism’ theory to biology. They show that mechanistic explanation by protein synthesis and natural selection are more closely analogous than they appear – both possess all three of these core elements of a mechanism widely recognized in the mechanisms literature. So that I think that mechanism theory is a good way in order to axiomatize population science, in inferring the formal structure which is implied in the properties of the scientific object of population science: fertility, mortality and migration. However, it seems to me to be important to axiomatize population sciences independently of biology as, while they have a common foundation, their scientific object is not the same. If we are now able to have a mechanistic explanation of population biology, we are still unable to know the basic combination of fertility, mortality and migration which will make possible the basic form of quantitative transformations in any given population.

Dear Jon,

I don’t find your response irritating and pointless. On the contrary, I find it interesting on an important subject. However I will try to tell you why I don’t agree with some of your points.

First, I think that you are using, for formal disciplines, a definition of axioms as postulates or assumptions, which is similar to the one used from the end of the XIXth century to recently. For example, Poincaré, in Science and hypothesis (1902), said that axioms are arbitrary conventions and that none of the various Euclidean or non-Euclidean geometries is truer than another. I don’t agree with such a definition which is contradicted by the fact that Euclidean geometry had been inducted from observation, and has not been arbitrarily chosen. Evidently new observations may induce a change in these axioms, such as the Einsteinian theory, but they are always linked to observed properties of the world. Even recent research had shown that the geometry of the whole universe is Euclidean (see Planck Hfi). So that, these axioms are not arbitrary conventions. The ideas of Bacon, Descartes and Newton are more convincing: they told us that axioms are inferred from the properties of a given system (here the universe). This notion of classical induction introduced by Bacon, which is different from the more familiar neo-positivist notion of generalisation from particular instances established by Locke and Hume, consists in discovering a system’s principle from a study of its properties.

Second, I don’t think that there is a real difference in constructing an applied discipline from constructing a formal discipline. Evidently the axioms will be different, but the methods to find the axioms and the methods to confirm their validity will be similar. I think that you can find a perfect example of such a construction in the book from your compatriot Deirdre Pratt, published in our Methodos Series: Modelling written communication (2011).

I will be very happy to have your reaction to these ideas.

With my best regards,

Daniel

Dear Jon,

Yes, I will be very interested to continue this quest with you, and I hope that some of our disagreements may prove to be fruitful for both of us.

First, I entirely agree that we have differences in definition of terms such as “axioms” and I will try to see why. You seem to agree with the definitions given by Wikipedia: “As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy… As used in modern logic, an axiom is simply a premise or starting point for reasoning”, even if you don’t present this article as authoritative. For me, this anonymous modern encyclopaedia is very often written by authors that have only a partial view on their subject, even if it may be corrected by other anonymous authors. What a difference with our French encyclopaedia of the 18th century with most of the papers authored by Diderot, D’Alembert, Voltaire, etc. As their views are linked to their published work, you can delineate them clearly, contrary to what happens in Wikipedia. Finally, there are many definitions of a given concept, and I want to see these different definitions to make a clearer idea of this concept and fix my ideas.

For the word “axiom”, I think that the definitions given by Wikipedia are fundamentally incorrect, even if they had been almost universally taught. If you are interested, I may come back later to formal Euclidean geometry about which I read a very interesting description of Pappus on analysis and synthesis in a volume on “The method of analysis” written by Hintikka and Remes (1974). I will just here tell you that Descartes told us that the Euclidean axioms are not premises without demonstration or starting point for reasoning, but that they had been demonstrated, evidently not by deduction, but by the analysis of properties of geometric abstract figures. The method that permits their demonstration has been called induction by Bacon: it consists of the deduction of principles from the study of their consequences. Latter this classical induction had been rejected by empiricist philosophy (Hume, Mill), which revived the traditional meaning of induction as generalisation and restored the deductive conception of explanation. Read Robert Franck book on “The explanatory power of models” (2002), for more details on this history.

I will not go further today, but I think that I will have later some more comments on your last response in three parts.

With my best regards,

Daniel

Hi Jon,

Yes, I agree that our discussion is going out of the field of population science, even if this point is important for them, and would be better in the field of philosophy of science. However, I will be very interested to continue it with you. Evidently by mail it will be easier, but I think it would be interesting to have the reactions of other Research Gate researchers. As we have the possibility to add attachments to our messages I think that this may be a good way to continue our exchanges. Would you agree with that? Even we can put a more general question on axiomatics in the Philosophy of science section of the kind: is an axiom simply a premise or starting point of reasoning, as Wikipedia said?

With my best regards,

Daniel

Yes, MS word documents are acceptable. As a first attachment, you will find what I wrote in my book on Probability and social science (2012) about the axiomatisation of objective probability. I undertook in this volume the same task for subjective probability and logical probability.

Your comments and the formulation of your more general concepts in this field will be welcome.

Best regards,

Daniel

Dear Jon,

Than you for your reflexions about the differences between physics and demography, and your thoughts about an axiomatic system designed for social sciences.

I entirely agree with you that the terminology used in physics should be used with great caution in demography, and in fact we don’t use such terms like “phase”, “phase change”, “symmetry” or symmetry breaking”. Even if we use some terms like “gravity models of migration”, “law of mortality”, etc., we are more interested by the “probabilities” (objectivist or Bayesian) of the different demographic phenomenon as mortality, fertility, migration, etc.

For the differences between two classes in a school multilevel analysis proved very efficient in order to disentangle this problem, analysing simultaneously the different levels; children, classes and schools (see the introduction of Harvey Goldstein book on Multilevel statistical models, 1995).

I agree also that there is no axiomatic basis for population sciences. As I said we had a number of paradigms: cross-sectional, longitudinal, event-history, multilevel. Each of them took the shortcomings of its predecessor as a starting point and offers a method for surmounting them, without however erasing the knowledge attained through earlier paradigm. However some non-linear cumulativity exists and the following sentence from Granger (1994) explains clearly what occurs: True, the human fact can indeed be scientifically understood only through multiple angles of vision, but on condition that we discover the controllable operation that uses these angles to recreate the fact stereoscopically.

With my best regards,

Daniel

Dear Jon,

I am happy to continue with you this discussion about an axiomatic basis for social sciences and demography.

In fact, we can find a great isomorphism between physics and demography when considering Lotka’s results (1939). He showed that when we consider a population with continuously changing conditions of fertility and mortality, this population follows a process of demographic change (P). Similarly when Newton considered a material point with continuously changing forces acting on it, this point follows in space a curve (C). If at a given time t, you stop the changes in fertility and mortality letting them remain constant in the future, then the demographic process will tend towards a stable population, keeping the same age structure, corresponding to the fertility and mortality conditions at time t. Similarly, if at time t you stop the changes in the forces, then the point will continue its move on the line tangent to the curve (C). There is only one difference: the stable population is not attained immediately, while the point follows immediately its tangent. However in some cases the stable population may be attained immediately, leading to semi-stable populations.

We can also consider as a symmetry the difference between births and deaths, the first introducing new elements in the population, the second subtracting others. The same is true for immigration and emigration. A phase change may be seen in the formation of a couple, with symmetry between man and woman, and permitting the birth of children.

What do you think about these preliminary ideas following yours?

With best regards,

Daniel

Reference:

Lotka, A.J. 1939, Théorie analytique des associations biologiques, Paris: Herman. (Translated in English 1998, Analytical theory of biological population, Springer).

Dear Jon,

Thank you very much for your previous comments on axiomatics in social science. The concept of an axiom as a basis for generalization in precision of the behaviour of a system in which entities of particular type interact in particular ways under particular circumstances is interesting, and I agree with you that axiomatizing the behaviour of human populations will be a difficult task. But however I think it may be possible, on simplifying the problem and may be on giving a more precise definition of an axiom in inductive science, as I will show later.

I begin to read your paper on intelligence in insect societies, and I appreciated very much the citation of Bacon you give at the beginning of first chapter. This led me to go to his definition of inductive science:

‘There are and can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgement and to the discovery of middle axioms. And this way is now in fashion. The other derives from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms at last of all. This is the true way, but as yet untried.’

This definition is very important as it distinguishes what he calls later the four idols (Idols of the Tribe, Idols of the Cave, Idols of the Market Place and Idols of the Theater) –errors of thinking and judgement- which for him besets men’s mind, from a true science. To apply the second way in social science, it is necessary to overcome the complexity of social life in limiting the social property under study, and restricting the number of characteristics to be involved. You see that here again I am rejoining your proposal of suitably small models, but may be more general ones.

That is all for the moment …

With my best regards,

Daniel