David Hilbert on Unification

At the end of David Hilbert’s Mathematical Problems, Hilbert goes into the details for his motivations for what we may call unity of science thesis. These reasons are as poignant today in my view as they were in his own time. Motvations could be summarised thusly:

 

  • Divergences/fracturing mathematics into subdisciplines will mean specialised areas will not engage with other areas outside their specialism

  • The most important innovations are driven by simplicity, more refined tools and less complication.

 

The first thesis is a problematic of overspecialisation and genrefication of any kind of academic research. Becoming so niche that one is essentially writing for a peer group that is too specific and few. Perhaps this is inevitable in the world of industrial research and constant innovation. If we are to believe that subdisciplines and specialisation are a necessity, then we cannot understand Hilbert’s second thesis, of parsinomy. Granted, more needs to be elaborated if such a unification thesis were to work. Unification has its own problems, but there is a bonus to clarity and it is a matter of fact that many great scientific innovations are of the sort that unify and simplify seemingly irrelevant areas (Maxwell Equations or Relativity for example).

 The conclusion of Hilbert’s lecture is as follows:

The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly.

But, we ask, with the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and cast aside older more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science.

The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples! [David Hilbert, 1900]

 


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The antiomies of the foundations

There is a distinct contradiction, and yet, agreement, in the following two propositions:

P1. Mathematics cannot be shown to be complete
P2. We cannot but conceive of Mathematics, properly construed, as ideally composed of a set of axioms such that all and any system of mathematics can be reduced to a common simple system, or set of axioms such that shows a common genus to all mathematics.

This view, I maintain, is a Kantian view of mathematics. Kant’s constraints upon the proper conduct of science is that there ultimately originates a primary concept, but, that this concept is knowable or discoverable, or even actual, is not relevant, nor should we be too concerned if we never find it.

For science to be proper, Kant says, it must fit an ideal of knowledge, but such an ideal is projected (this entails the ideality of natural kinds) and not real. Such an ideal also seems to suggest that we use a bit of elipsis in our explanations and descriptions of science. A Kantian view of science also would set as a desideratum that there were a formalisability/mathematicisation constraint on anything if it is to be proper science at all.

The ideal is a projection, and is an “as if it were real” constraint (that is the ellipsis to which I speak of). Because it is a projection, our kinds and entities and laws within the scientific frame work not only can be subject to change, but desirably so, are they changeable, for scientific theories could always change, and are not rigidly set.

Rigidity is still present in the Kantian conception of science, however, in the desideratum of the constructability of formal langauges upon which we describe our phenomena. Consider the difference between ‘Water’ (h20) and water (that stuff we drink). Most, if not all the water we come across is not ‘water’, perhaps in some ways, ‘water’ does not exist, HOWEVER. Water necessarily presupposes ‘water’, in virtue of its ideality. For what makes water1 the same as water2 other than h20? Nothing.

H20 is criterial of water, but in a way, its pure form is never to be found in water, only ‘water’, which projects onto all thigns called water, makes sense of our empirical concept in such a way to be science. But, because ‘water’ is a priori regulatively ideal, it is also subject to change. The contradiction is, then, how is water necessarily h20, yet only indexical to our scientific understanding?

The answer to this lies in the conception of necessity. Necessity here, is defined as a criterial relation. Therefore, to say that “2 is a number” is necessarily true is to state a criteria. Necessity is criteria. But then, is not necessity similar to possibility? For criteria presupposes the conditions, and conditions is construed in the Kantian system as possibility. It would seem then that necessity can only take place as a concept where possibility is first defined, such that in a sense, necessity is only possible if, possibility allows, and this is necessarily so.

Destre (and Michael)