Continuing my reply to David, started here. I’m going to be getting into mathematical conceivability, which David spent much more time on.

After first noting that the section wherein I make the positive argument in favor of my thesis is “fascinating” and “impressive,” as if he seems to agree, he appears to take this back in outlining several problems my thesis has when it comes to the realm of mathematical propositions. My thesis applied to mathematical propositions is this;

To conceive of a logico-mathematical proposition is to know that it is true; the demonstration of its conceivability is the same as the demonstration of its actuality.

This follows by simple logic. Assuming that whatever is conceivable is possible, then it cannot be the case that false mathematical propositions are conceivable, since they are impossible. Hence, the conceiving of a mathematical proposition is essentially tied to the way in which it is known to be true.

David brings up the obvious problem, that it seems we cannot first have an idea of what is to be proved. If we are so unable to conceive a theorem before its proven true, then what is the mathematician doing when they set out to prove a not-yet-proved theorem? They can hold in their mind *a* condition of that theorem’s being true, that the theorem (expressed mathematically) will follow from other given truths which are already known or can be known. Thus it seems that the mathematician has some conception of the theorem, even if it turns out false upon further analysis.

To help ground out why this shouldn’t be taken as problematic I developed a parallel case of reasoning in this previous post. If we consider the sudoku, we will notice that there are;

1) rules of the game which imply a unique possibility of every rule-rigorous sudoku 2) the necessity of placement of numbers provided certain premises 3) our ability to postulate the placement of numbers under present conditions of lack of knowledgeNow it is taken that for a sudoku which exists a unique solution (otherwise it just isn’t a proper sudoku) from the starting numbers, we might postulate that a 9 inhabits some particular square. We can even postulate *a* condition of its being the case; that it doesn’t lead to absurdity with other inferentially known squares. However, is this to conceive that it actually inhabits that particular square by the rules of the sudoku? Not precisely. Granted, it is conceived in a mode, but it is conceived as something like this; “9 in the case that it isn’t absurd, which we’ll find out.” This is not the same as conceiving the 9 to be actually in that square, under which our conception would be in a direct mode, “9 because it is inferred to do so by the rules of the game.”

I propose that what the mathematician is doing is similar. When the mathematician considers some unproved (to themselves) theorem, they conceive it by some mode, but by a mode which is different from conceiving the proposition in itself. For instance, they might conceive Goldbach’s conjecture as “true in the case that it isn’t absurd, which we’ll find out (or maybe not).” This is not the same as conceiving it in its own right, as “true because it is inferred to do so by the rules of logic-math.”

This is a somewhat lengthy pre-emptive strike on the tack that David takes towards my thesis in regards to mathematical conceivability, and I believe it successfully obviates his concerns. There is no essential premise of formalism in my account. It doesn’t require postulating that mathematics exists as some unchanging body of knowledge for which new axioms and systems aren’t worked out.

The only other point worth taking up is David’s treatment of unprovable theorems. My thesis doesn’t give a commitment to the provability of every true theorem. To conceive of an unprovable theorem is like conceiving of a not-yet-proved theorem; “true in the case it isn’t absurd, which we’ll find out (or not).” It is conceived by a mode, which is not to conceive it in-itself. Hence my thesis is perfectly compatible with Gödel.

This being the case, I don’t need to accept any of David’s provisions to save my thesis. At the very least, it did help to tackle mathematical conceivability, as now I have the notion of “conceiving by a mode.”

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