Walicki's opinion concerning my lecture in the theory of science

Walicki's opinion concerning my lecture in the theory of science

John Bjarne Grover

As to "Kuhn's paradigms and the simultaneity of scientific discoveries", by John Grøver.

I was asked to give an expert opinion relative to the more specific mathematical and logical aspects of the lecture. I therefore look apart from what looks like the 'main message', from approx. page 28 onwards, where the relationship between 'simultaneous discoveries' and 'paradigms' is transferred from a science-theoretic and -historical context to a somewhat more private and esotherical sphere.

I would sum up the contents of the first 10 pages as follows:

1. Introduction of the concepts of 'simultaneous discoveries' and 'paradigma' ("shared consciousness").

2. Indication that "computability" can be considered as a paradigma in the period 1900-1960, along with a distinction between
2a. "Turing-computability" vs.
2b. "Cantor-computability"

3. Discussion of the three factors which Kuhn recognize in 'simultaneous discoveries', with the main thesis that such processes emerge in a process of 'rationalization' of "network of converging but disparate scientific disciplines" (which seem to refer to 'shared consciousness').

4. Mathematics is supposed to be the most 'rationalizing' science, and a widespread presence of 'mathematical platonism' shall inicate that 'shared consciousness' is an essential part of mathematics.

5. Therefore, one should easily find 'simultaneous discoveries' in mathematics, and a series of three classic cases follows.

6. The seemingly evident - but unpronounced - thesis, viz., that 'shared consciousness' is a basis for 'simultaneous discoveries', receives an inter-pretation in the discussion of "the concept work" and the alleged ambiguity in this expression such as it is used by Kuhn.

7. The main thesis, on page 25, says that "the revolutionary work with mathematical concepts is also revolutionarily present in the platonic regions of the shared consciousness, which consequently means that the work can be perceived by other mathematicians..."

Let me briefly comment on some of the points:

7.
Since the concept of "the platonic regions of the shared consciousness" remains entirely unexplained and unspecified, it is difficult to say what should be the author's contribution in clarifying the relation between the 'paradigms' and the 'simul-taneous discoveries' in mathematics (and elsewhere). The author does, though, indicate a possible influence on individuals through this sphere, but this influence has a character of completely "irrational powers at work". It is furthermore asserted (page 25f.) that a paradigm change in geometry should be implemented "in the form of a simultaneous discovery with contribution from at least two mathematicians". The implied necessity of 'simultaneity' suits the author's few examples well, but it does not suit well to a series of other situations which also could be called "paradigm shifts", wherein no 'simultaneous discovery' took place (for example the discovery/introduction of Euclidean geometry, irrational numbers, Galois theory, Cantor's transfinite numbers, Frege's logic, etc.).

6.
The paragraph on 'the concept work' was almost entirely incomprehensible to me and I could not determine it relative to the rest of the argument. The quote which should illustrate the difference between two alleged interpretational alternatives illustrates nothing.

2.
The presentation of the basic thoughts and results of logic reveals no fundamental deficiencies. There are only a few minor things which remain unclarified:

i) On page 12, it is asserted that "in order to interpret the new computers, the boundary to the knowledge which was to count as computable was defined ... at the turn of the century". At this time, nobody had as yet any ideas of "the new computers", so unless the author here wants to indicate some kind of developmental determinism a la Hegel or Marx, it sounds a little strange.

ii) Furthermore on page 12: "it was Cantor's explicit intention to study the theological interpretation of mathematics". One should be aware that Cantor excluded Russell's paradox (before it was discovered) precisely by not mixing mathematics and theology - according to Cantor, the universe of all objects could not be considered a mathematical object, because this kind of universe would be a potential object for God but not for mathematics.

iii) Page 13: "...were made in the thirties. Tarski defined the new semantics, the new computability boundary was defined, and the computer was developed as the technical tool to handle this new knowledge...". If by "new semantics" defined by Tarski is meant "formal semantics", then this semantics has little to do with the definition of computability. 'Computer' was not developed "in the thirties".

iv) Page 14: "...anything which can be transmitted from one scientist to another ... is also definable in the form of a Turing machine. This is today generally acknowledged in the practice of computer implementation...". This sounds directly insensible to me. People who work with (theoretical) informatics are perhaps particularly aware of the obvious limitations in computers.

These and a couple of other places indicated for me that the use of terms such a "computability", "Turing-machine" etc. is not always to be understood literally in a technical sense, but that they function more as labels for more general phenomena which the author tries to approach. In this connection, there also comes the distinction between
2a. "Turing machine" and
2b. "Cantor machine"
which is fairly unclarified since 2a cannot be understood as a technical concept. In an honest attempt, I have strived to interpret 2a. as 'normal science', what is 'inside the paradigm', and 2b. as a kind of metalevel, or precisely a potential source for paradigm shifts. But, e.g., the example on pages 14-15 indicates that 2a. nevertheless should be interpreted in a technical sense. It would be helpful with a somewhat more clarified exposition of the concepts and distinctions one works with.

My general impression is that the author has satisfying knowledge of existing mathematical concepts and historical examples. To the extent that one can talk of presentation of any original ideas, these are at best obscure and inaccessible.

Michal Walicki
senior scientific officer