On 2016-11-02 21:15, Marta Bunge wrote:

... much of mathematics has been inspired by a desire to apply it to the natural sciences.

Yes, even if indeed

... "much" and not "all",

but to me, the driving force concerning that

... freedom which allows the invention of objects such as the complex numbers or of the infinitesimals.

is unclear because it has been made part of that freedom.

Now, what about actual applications of science? Those are not just the product of the theories themselves, but also of experimentation and of successive approximations. It is for this reason that I see no mystery in that certain scientific theories can sometimes be succesfully applied.

Many, even if again not all, mathematicians like it when application fits their theory, but they are less interested in fitting their theories to applications. Applying mathematics in natural science and from there apply in the world, mostly then means the geometric world, the measurable things. Applying mathematics for the world of judgment and assessment, and personalization, like when using terminology in health and social care, is more tricky, and unfortunately, very few mathematics engage in such things. This is, by the way, the reason why "evidence" in health is still just about comparison of mean values over populations. Treatment must be personalized and medicine do not have a formal solution for that. My posting now is a bit related to my previous posting about meta and object, and about what comes first. When we try to deal with medical terminology we enter algebraic and logic machineries, but if not invoking categories means everything is over Set. Clearly, and if we take the simple example like the interplay between disorder and functioning, both of them have structure, and they are different. But they affect each other. Can we map disorder to functioning, or does disorder act on functioning (which, by the way, is not the same as functioning acting on disorder). What if we include drug ontology, or treatment ontology more in general. Non-commutativity, modules and monoidal categories could be useful. Maybe, maybe not. When such questions arise, we see how they arise from the application side. This brings professional language into the picture and how to deal with it in relation to mathematical language. Some of it is a question about syntax, other parts are semantic, but in different ways. Here I would say we have virgin areas for investigation, in particular for the CAT community, e.g. as compared with the FOM community. I want to underline, that if we apply, we must fit theory to application, not the other way around. Or said in another way, the world should disrupt math so that math can continue to disrupt the world. This to me is the special role of math among other sciences. Best, Patrik PS Why not also try something out about the empty type. The only thing in mathematics that axiomatically is, not just exists, but is, is the empty set, and then we put brackets around, and create natural numbers. From natural numbers we then create everything else in mathematics. Couldn't the empty type be something similar? But we shouldn't lean on the HoTT community because they don't comply with mathematics in their HoTTematics. They are two dofferent worlds. Set theory is untyped, so math is weak on types. On 2016-11-02 21:15, Marta Bunge wrote:

Dear Andre,

Of course I agree with you in that the natural world is only partially explained by science. I also believe that much of mathematics has been inspired by a desire to apply it to the natural sciences. Notice, however, that I said "much" and not "all", and here is where mathematics and the natural sciences differ. Mathematicians have a freedom not afforded to scientists. It is this freedom which allows the invention of objects such as the complex numbers or of the infinitesimals. Now, is it a mystery that such products of the human mind find applications in scientific theories, or is it rather that the latter themselves are also the product of the human mind? After all, it is only through rational thinking (including intuition) that we are able to (believe we) understand the natural world. Now, what about actual applications of science? Those are not just the product of the theories themselves, but also of experimentation and of successive approximations. It is for this reason that I see no mystery in that certain scientific theories can sometimes be succesfully applied. Whether this is or is not a metaphysical point of view it is not for me to say.

Best regards,

Marta

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