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Abstraction: Einstein on Mathematics+Theory+Reality
Technorati Tags: Albert Einstein, physics, formalized theories, interpretations of theories, reality, abstraction, models of theories, empiricism
[update 2010-02-23T19:46Z I had to change the title of this post. My narration below revealed to me that theory is the statement (a claim) that connects the logical-forma mathematics to features of reality, and is therefore the statement of the theory that posits the validity of interpretation that connects to a model in reality (not of reality, so far). I’m also fixing a typo while I’m here.]
The abstraction of formal theories away from any appeal to nature is a theme for me. I claim that it gives us a powerful way to appreciate nature. I find it indispensible in teasing out what makes computers useful and what part software developers play in having that work for us. To my great pleasure, this is not a new or particularly radical notion at all, although relatively new to me. Albert Einstein had his own appreciation.
“It is mathematics which affords the exact
In 1921, Albert Einstein, then 42, gave an address in which he explained what was necessary for him to appreciate about the difference between mathematics and theories about the physical universe in order to formulate the theory of relativity. He began by pointing out a peculiar situation around the applicability of mathematics to practical affairs.
For Einstein, this is not a question; he will answer the second question in the negative. His analysis of the first question is foretold with this response:
He gives credit for clarity on the matter to the introduction of mathematical logic or “Axiomatics,” put this way:
At the time that Einstein speaks of this as such a powerful innovation, Whitehead and Russell’s Principia Mathematica has only been in print since 1910. David Hilbert’s famous 23 problems had just been announced in 1900 when Hilbert had already made a formal axiomatization of Geometry and demonstrated its completeness. In contrast, Kurt Gödel was about 15 when Einstein gave this talk. Alan Turing was 9. (My mother was 4.)
Einstein uses Geometry to illustrate the separation between the logical-formal mathematical expression of a theory in what he termed the modern formulation:
This completes the separation of mathematics, expressed as applications of mathematical logic, establishing the separation of mathematics from any reference to reality:
So how is the appropriateness of mathematics to real-world matters accomplished? Einstein sees it this way:
Einstein prescribes a way, via necessarily-informal language, by which geometry can be interpreted to apply to aspects of the natural world with it asserted (the theory, in this matter) that the logico-formal conclusions of geometry are valid conclusions about the natural world. Verification of that assertion depends on experience, not anything expressed in the formal geometry.
It is in this sense that I mean Reality is the Model. Einstein suggests that one can make a theory in physics by asserting this correspondence, an interpretation, between logico-formal geometry and the natural world. For Einstein, this practical geometry (or perhaps better, physical geometry) is a completion of logic-formal geometry. I prefer to have sharp separation with physical geometry as one distinct model of formal geometry. This makes room for yet other models not tied to natural objects and essentially independent of each other. (Consider the virtual geometry of worlds that exist only in images, such as the 3D experience of Avatar.)
However applied, the critical feature of the separation of mathematical (or mathematical-logico) formulations from interpretations in nature or elsewhere is the unexpected power it provides. I shall leave Einstein with the final word:
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