Well, you can multiply the ('imaginary') square root of minus 1 by the square root of minus 1 and (according to some rules) get 1. Though not in real numbers as I understand it. Which is only partially. I'm not even sure it's allowed/valid in imaginary numbers or complex numbers. The analogy with this in the real world would be a bit like saying that you can get a possible from combining two impossibles. Quite a feat. But, as I see it (and I'm not really really a mathematician, I just tutor it up to A level and even at that I'm not fully up to speed on A2 yet) the internal rules are ultimately arbitrary.[/QUOTE]subsymbolic;672780 wrote:I think we left the ladder behind us when the Principia Mathematica was published it just helped us to discover maths but led us astray beyond that. There's been a fair bit done since then and maths has it's own unique world. Either way, you are the mathematician and I most certainly am not. So have a think about whether there are any things that can be done by following the laws of maths but not by following the laws of physics. Because if there are then as I said, we have a choice of axioms...ruby sparks;672779 wrote:Oh I'm not saying that there's no such thing as deduction. I just tend to think it's all ultimately predicated on experience, if you dig down far enough, to the foundations. And I tend to think of experience as only providing inductive justification. As you say, we leave a ladder behind.subsymbolic;672777 wrote:

Sorry added a bit in a cross edit:

Now I really must get going.

Arbitrary as a whole perhaps, but once you are playing the tautological(?) game called maths, the rules are pretty damn fixed. Although you do get a choice of axioms as I keep hinting.

So if I were being stalked by Laplace and he were predicting my every move using physics, would I be able to evade him if I were to decide to go left if I could make an impossible out of a possible and right if I couldn't? There are better examples, but this will do.

It's a minimal case, but it does demonstrate some of the teeth of stances and anomalous monism in a way that should be more palatable. Imagine if you had a case like this but involving an absolutely false theory that allowed you to predict your own behaviour and act on the prediction. Not as tidy as Banach Tarski, but it's still objects that don't and can't exist in a physical realm!