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After thinking more about: Daniel Dennett's concept of free will as an equation of state?

I am super confused about the linguistics concerning mathematics. For example, "take the limit of x to 0 and then then limit of y to infinity; else the equation will diverge"

Notice the subtleties:

Firstly, it is natural for us to put something like a notion of time in our language (then). Secondly, our style of doing math assumes we can think of math as an equation of state with degrees of freedom possibly because of Dennett's freewill (see the first link).

Daniel Dennett's concept of free will, in which he argues that our choices are the result of complex computations that take into account our desires, beliefs, and goals, as well as the external world. I realized that the nature of these computations would not be modeled by what physicists call a time evolution equation but, rather, an equation of state.*

*To elaborate a bit more: Consider the ideal gas law - PV = NRT. I understand the relationship between pressure and, let's say, temperature. Notice that I do not know the time evolution of the system. However, I can say, "If I increase the pressure, the temperature will change." Regardless of whether a time evolution equation for "me" + "gas in a box" exists (or whether the degree of freedom of "if" exists), I can perform this computation.

**Note: When I say "as an equation of state" I'm not thinking the mind is actually putting in numbers but rather it is doing an activity which can be mapped to that. Consider this example: I am asked the time signature of a pop song. I can do so by merely headbanging and instinctively knowing it's 4/4 or I can count. Both activities can be mapped to same output and inputs.

Math is an abstraction! One which can be abstracted a particular way or does our notion of freewill obstruct that of Plato's heaven we can access? In other words is there an extension of mathematics we cannot access because of our notion of will?

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  • Based on the sci-fi: youtu.be/tFMo3UJ4B4g?si=g61NJb-xl9dA3JAW :p Oct 30, 2023 at 5:29
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    ""take the limit of x to 0 and then then limit of y..." means "For every x "sufficiently close" to 0 etc" nothing to do with free will. Oct 30, 2023 at 7:31
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    Plato noticed it too:"They speak, I suppose, very laughably and perforce, for they mention squaring, applying, and adding, and state all their claims as if they are engaged in action and fashioning all their proofs for the sake of action..." Linguistics of informal math is affected by whatever folk intuitions affect the language generally, "free will" included. This is why mathematicians worked hard, since before Plato, to make sure that those are only used where they speed up comprehension but do not affect substance. Façon de parler cannot tell us what we can or cannot access.
    – Conifold
    Oct 30, 2023 at 7:47
  • @MauroALLEGRANZA but our interaction with this abstraction is through our first world perspective interaction of freewill which affects our language and in turn thinking Oct 30, 2023 at 7:52
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    Seeing as this question is dear to my heart (no cap!), I will indicate reflection on Brouwer's free choice sequences first, and later today I will shoot for a sustained response. Oct 30, 2023 at 11:15

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If we take intuitionism as the paradigmatic example of a philosophy of mathematics in which free will plays a constructive role, via the free choice sequences unfolded by Freely Creating Mathematical Subjects, then per intuitionism's usual rejection of actual infinity, it might seem to us that appealing to the possibilities of the will to guide our mathematical practice will limit us from using concepts of things like large cardinals or (∞, n)-categories. (How actual the infinity of those categories is, is not clear to me, however; on the other hand, large cardinals are meant as strong examples of actual infinity.)

However, if we have the will-to-refer, and can use this to will to refer to whatever we can conceive of, then why would we not be able to will to refer to actually infinite things? In fact, it is not even entirely clear that the most sensible such distinction is the actual/potential one: per Cantor, we have the relatively and the absolutely infinite instead, and absolute infinity replaces actual infinity as the unattainable possible referent of our will-to-meaning. That is, the supremacy of absolute infinity is also an epistemic barrier for us, akin to what the intuitionist thinks about so-called actual infinity.

For all that, then, if we imagine forcing, for example, as an action of our minds, and if we go to the modal logic of forcing and append a thesis that mathematical free will really is some sort of "ability to think otherwise" through a set of alternative possibilities, then forcing, with all its tenebrous and wistful power, becomes an icon of free will to this extent. Or, then, the set-theoretic multiverse standpoint exemplifies our will, here, for it is by our will to work in whichever set world that we traverse this or that multiverse, is it not?

Consider this excerpt from Zoble[08]:

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What is this talk of "shooting a club"? It sounds like an action, again. Yet it is a metaphorical action pertaining to highly elaborate set-theoretic "machinery," so one wonders whether the metaphor implies a weakness or obstacle in our mathematical abilities?

I would also recommend looking through Cheng[04]. She discusses how a lot of mathematicians, especially category theorists, use phrases like "morally true" or "ought to be true" in their work. She eschews an appeal to mathematical agency, granted, to some extent, but it is not very difficult to imagine that there might be a way to extrapolate the metaphor of the will-to-refer and reference-as-a-free-action from the given phraseology.

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