[FOM] Unreasonable effectiveness
Sam Sanders
sasander at cage.ugent.be
Tue Nov 5 12:19:25 EST 2013
Dear FOM-readers,
One point of view or aspect regarding "unreasonable effectiveness" has perhaps not been addressed, namely robustness.
There is a lot of informal mathematical reasoning in physics, often involving an intuitive infinitesimal calculus. While this practice
would in general produce false results (see Cauchy's well-known mistake), for the specific purpose of physics (modeling and predicting the physical word) this practice
works surprisingly (unreasonably?) well. Somehow, the math used in physics is very "fault-tolerant".
Idealization is another aspect of physics (and the exact sciences). Again, there is a part of mathematics which can accommodate
this practice, i.e. small errors do not necessarily blow up and make the results meaningless.
I would go as far as claiming that any "unreasonable effectiveness" is in large part due to the robustness of the mathematics
used in physics: There are countless way errors can creep in, but somehow the math still produces meaningful results due to its robustness.
By contrast, e.g. in philosophy, arguments are much more non-robust in that they often amount to splitting hairs .
(This is the naive view of a mathematician/physicist for the sake of argument and is not meant to belittle philosophy or its value).
Another example is art or literature where a dozen critics can have thirteen opinions on "what the artist meant with this piece of art".
For these reasons, I do not think an account of physics can only be about certain "structures" and the mathematics on those structures,
as these structures are (highly likely) only approximately realized in reality (in answer to the below email). One has to consider structures, the mathematics on those
structures, and an account of how well the math still works if one deviates (slightly) from said structures, i.e robustness.
Perhaps one can rephrase "unreasonable effectiveness" in terms of robustness: "How is it that mathematics produces such
robust tools?" (as opposed to other human endavours).
Best,
Sam Sanders
On 04 Nov 2013, at 20:23, Steven Ericsson-Zenith <steven at iase.us> wrote:
> This naturally begs the question of what it is to say, in exact terms, that mathematics is "unreasonably effective." What would an answer to the question look like? It is a general answer or specific to one or more mathematical domains?
>
> My own view is that we do not have the means currently, in any form of mathematical logic (computerized or not), to address the question. But you might begin with the end cases and they may point the way. These are not philosophical questions but necessary existential questions in relation to the physical sciences: I would not take the axiomatic method for granted, for example, without asking what it is that binds (or selects) the axioms. The other end case to consider is how you demonstrate "unreasonable effectiveness" in the face of necessary fallibility in the physical sciences. This pretty much brings you back to reasoning in terms of known properties of mathematical systems.
>
> The physical sciences has no clearly articulated consensus on epistemology right now, so I do not know where you would begin, except to address the epistemic issues.
>
> I will be proposing the following in a Stanford University lecture next week: if you chose Einstein's advocacy of general covariance as the basis of all physical law (that, I argue, is to be preferred to conventional axiomatic systems since it provides the lost binding between premises) and a form of structuralism as the basis of the physical sciences, and you take structuralism to be the subject of pure mathematics, then you will begin to get traction on your question - at least then pure mathematics and the physical sciences have a common subject, structure. But I do not expect that such an approach will initially be welcomed by contemporary theoretical physics.
>
> More controversial, in terms of mathematical logic, I believe that the way you build the bridge between pure mathematics and the physical sciences is by building on the above; that is, the common subject for mathematical logic must also be a structuralism. In historical terms, I think that Benjamin Peirce's intuition, mentioned earlier, concerning the potential of complex analysis was good.
>
> Regards,
> Steven
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