I guess there are my beliefs-which-predict-my-expectations and my aliefs-which-still-weird-me-out. In the sense of beliefs which predict my expectation, I would say the following about mathematics: as far as logic is concerned, I have seen (with my eyes, connected to neurons, and so on) the proof that from P&-P anything follows, and since I do want to distinguish “truth” from “falsehood”, I view it as (unless I made a mistake in the proof of P&-P->Q, which I view as highly unlikely—an easy million-to-one against) as false. Anything which leads me to P&-P, therefore, I see as false, conditional on the possibility I made a mistake in the proof (or not noticed a mistake someone else made). Since I have a proof from “2+2=3” to “2+2=3 and 2+2!=3″ (which is fairly simple, and I checked multiple times) I view 2+2=3 as equally unlikely. That’s surely entanglement with the world—I manipulated symbols written by a physical pen on a physical paper, and at each stage, the line following obeyed a relationship with the line before it. My belief that “there is some truth”, I guess, can be called unconditional—nothing I see will convince me otherwise. But I’m not even certain I can conceive of a world without truth, while I can conceive of a world, sadly, where there are mistakes in my proofs :)
You’re missing the essential point about deductives, which is this:
Changing the substrate used for the calculations does not change the experiment.
With a normal experiment, if you repeat my experiment it’s possible that your apparatus differs from mine in a way which (unbeknownst to either of us) is salient and affects the outcome.
With mathematical deduction, if our results disagree, (at least) one of us is simply wrong, it’s not “this datum is also valid but it’s data about a different set of conditions”, it’s “this datum contains an error in its derivation”. It is the same experiment, and the same computation, whether it is carried out on my brain, your brain, your brain using pen and paper as an external single-write store, theorem-prover software running on a Pentium, the same software running on an Athlon, different software in a different language running on a Babbage Analytical Engine… it’s still the same experiment. And a mistake in your proof really is a mistake, rather than the laws of mathematics having been momentarily false leading you to a false conclusion.
To quote the article, “Unconditional facts are not the same as unconditional beliefs.” Contrapositive: conditional beliefs are not the same as conditional facts.
The only way in which your calculation entangled with the world is in terms of the reliability of pen-and-paper single-write storage; that reliability is not contingent on what the true laws of mathematics are, so the bits that come from that are not bits you can usefully entangle with. The bits that you can obtain about the true laws of mathematics are bits produced by computation.
I guess there are my beliefs-which-predict-my-expectations and my aliefs-which-still-weird-me-out. In the sense of beliefs which predict my expectation, I would say the following about mathematics: as far as logic is concerned, I have seen (with my eyes, connected to neurons, and so on) the proof that from P&-P anything follows, and since I do want to distinguish “truth” from “falsehood”, I view it as (unless I made a mistake in the proof of P&-P->Q, which I view as highly unlikely—an easy million-to-one against) as false. Anything which leads me to P&-P, therefore, I see as false, conditional on the possibility I made a mistake in the proof (or not noticed a mistake someone else made). Since I have a proof from “2+2=3” to “2+2=3 and 2+2!=3″ (which is fairly simple, and I checked multiple times) I view 2+2=3 as equally unlikely. That’s surely entanglement with the world—I manipulated symbols written by a physical pen on a physical paper, and at each stage, the line following obeyed a relationship with the line before it. My belief that “there is some truth”, I guess, can be called unconditional—nothing I see will convince me otherwise. But I’m not even certain I can conceive of a world without truth, while I can conceive of a world, sadly, where there are mistakes in my proofs :)
You’re missing the essential point about deductives, which is this:
Changing the substrate used for the calculations does not change the experiment.
With a normal experiment, if you repeat my experiment it’s possible that your apparatus differs from mine in a way which (unbeknownst to either of us) is salient and affects the outcome.
With mathematical deduction, if our results disagree, (at least) one of us is simply wrong, it’s not “this datum is also valid but it’s data about a different set of conditions”, it’s “this datum contains an error in its derivation”. It is the same experiment, and the same computation, whether it is carried out on my brain, your brain, your brain using pen and paper as an external single-write store, theorem-prover software running on a Pentium, the same software running on an Athlon, different software in a different language running on a Babbage Analytical Engine… it’s still the same experiment. And a mistake in your proof really is a mistake, rather than the laws of mathematics having been momentarily false leading you to a false conclusion. To quote the article, “Unconditional facts are not the same as unconditional beliefs.” Contrapositive: conditional beliefs are not the same as conditional facts.
The only way in which your calculation entangled with the world is in terms of the reliability of pen-and-paper single-write storage; that reliability is not contingent on what the true laws of mathematics are, so the bits that come from that are not bits you can usefully entangle with. The bits that you can obtain about the true laws of mathematics are bits produced by computation.