In his Introduction to Mathematical Philosophy (p. 4) Bertrand Russell characterises this dynamic:
(…as) all terms that are defined are defined by means of other terms, it is clear that human knowledge must always be content to accept some terms as intelligible without definition, in order to have a starting-point for its definitions.
This resonates with the issue of key theoretical assumptions being perhaps only retrospectively apparent at that nexus of creative insight and revelatory catharsis of intellectual discovery. Axiomatic systems of thought tend to contain (at least) two salient features, namely: their extensivity and their tendency to possess blind spots. We inevitably must take some things for granted without full and complete definition but in scientific theory the proxy for this is that we may not necessarily be aware of which assumptions are incomplete, inaccurate or faulty; that is, which definitions are being unknowingly taken on faith. The axiomatic blind spots and faulty assumptions which may be inaccessible from within the intelligible logical matrix of a theory are also precisely where system extensivity can occur.
While it is arguable that a conceptual model of scientific discovery as being a purely recombinatory process of addition and growth is intuitive and fairly simple to communicate (or agree upon), it somewhat misses the mark. As itself being a relatively formal logical system of axiom, reference and meaning (and as much as science accumulates, incorporates and generates data, facts and theory) science is also a system in which progress is marked logically by the internal extensivity of that system. This is not so much a Tower of Babel as it is an instance of (a) Cantor’s Dust-like internal extensivity in which the process of structural addition and theoretical revelation is also definable as a logical process of reconfiguration and reorganisation more than literal additive growth.
The various assumptions and articles of suspended disbelief (or of rational faith) upon which definitions and axioms rest are perhaps only made clear at the paradigm-shift represented by significant insight and discovery. Shifting up from Newton to Einstein allows retrospective clarity concerning the nature of gravity and that the assumption of it being a force was a crucial blind spot, reflexively justifiable from within it the semantics of its own theory but nevertheless hinting at something grander in the measured anomalies in observations of planetary motion. The extension of the system of thought incorporates and rationalises the existing framework of explanation but explains anomaly or observational discontinuity by indicating and unravelling unacknowledged assumptions. Extensivity in this way of course introduces new conundrums and indicates other anomalies to be explained, themselves suggesting further faulty assumptions (oh, hello dark matter !).
I have clearly been influenced by Gödel’s insights at some point and have an admittedly limited understanding of the curlicues of mathematical physics but at an intuitively analytical level I have a question or two to put forward. The places in which we find anomaly, discontinuity and aberration between theory and reality appear to be the places in which some assumption, some fundamental building block of theory is misrepresented, miscalibrated or otherwise unintentionally displaced from (a) truth and prospective leap in the procedural refinement of theory or the generation of completely new explanatory systems. So, inverting (or intuitively extending by creative self-reference) this situation somewhat, is there some method of logical analysis (human or machine) through which observational anomalies and their associated (axiomatic) definitions can be inserted back into the theoretical (logical) frameworks in which they appear such as to indicate, if not how assumptions are incorrect, at least which assumptions are key targets for reassessment ? Are there purely logical ways to indicate directions of research and theory ? It may be that the assumptions which really should be questioned are rarely those that actually are.
I am not suggesting that pure logic can solve theoretical conundrums; more that the nature of discovery and system extensibility takes certain kinds of shapes, forms, perhaps contours and relationships of self-reference and recombination or reconstellation of axioms and entities. Discontinuity indicates opportunity and there are very likely reproducible patterns, symmetries and insights at a purely abstract, logical layer of analysis in scientific theory which when highlighted may (or may not) assist the boffins in their work. What science is doing materially may not be directly indicative of what it is doing logically and there may (or may not) be a place for pure logic or a creative systems analysis in the discovery of new theories and scientific discovery.
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There is more to be said on this… there is always more to be said…