So what part of a mathematical universe do you find distasteful?
the idea that “2” exists as an abstract idea apart from any physical model
It’s this one.
Okay, but if actual infinities are allowed, then what defines small in the “made up of small parts”? Like, would tiny ghosts be okay because they’re “small”?
Given that you’re asking this question, I still haven’t been clear enough. I’ll try to explain it one last time. This time I’ll talk about Conway’s Game of Life and AI. The argument will carry over straightforwardly to physics and humans. (I know that Conway’s Game of life is made up of discrete cells, but I won’t be using that fact in the following argument.)
Suppose there is a Game of Life board which has an initial state which will simulate an AI. Hopefully it is inarguable that the AI’s behavior is entirely determined by the cell states and GoL rules.
Now suppose that as the game board evolves, the AI discovers Peano Arithmetic, derives “2 + 2 = 4”, and observes that this corresponds to what happens when it puts 2 apples in a bag that already contains 2 apples (there are apple-like things in the AI’s simulation). The fact that the AI derives “2 + 2 = 4″, and the fact that it observes a correspondence between this and the apples, has to be entirely determined by the rules of the Game of Life and the initial state.
In case this seems too simple and obvious so far and you’re wondering if you’re missing something, you’re probably not missing anything, this is meant to be simple and obvious.
If the AI notices how deep and intricate math is, how its many branches seem to be greatly interconnected with each other, and postulates that math is unreasonably effective. This also has to be caused entirely by the initial state and rules of the Game of Life. And if the Game of Life board is made up of sets embedded inside some model of set theory, or if it’s not embedded in anything and is just the only thing in all of existence, in either case nothing changes about the AI’s observations or actions and nothing ought to change about its predictions!
And if the existence or non-existence of something changes nothing about what it will observe, then using its existence to “explain” any of its observations is a contradiction in terms. This means that even its observation of the unreasonable effectiveness of math cannot be explained by the existence of a mathematical universe outside of the Game of Life board.
Connecting this back to what I was saying before, the “small parts” here are the cells of the Game of Life. You’ll note that it doesn’t matter if we replace the Game of Life by some other similar game where the board is a continuum. It also doesn’t even matter if the act of translating statements about the AI into statements about the board is uncomputable. All that matters is that the AI’s behavior is entirely determined by the “small parts”.
You might have noticed a loophole in this argument, in that even though the existence of math cannot change anything past the initial board state, if the board was embedded inside a model of set theory, then it would be that model which determined the initial state and rules. However, since the existence of math is compatible with every consistent set of rules and literally every initial board state, knowing this would also give no predictive power to the AI.
At best the AI could try to argue that being embedded inside a mathematical universe explains why the Game of Life rules are consistent. But then it would still be a mystery why the mathematical universe itself follows consistent rules, so in the end the AI would be left with just as many questions as it started with.
the idea that “2” exists as an abstract idea apart from any physical model
It’s this one.
Note that “Platonism false” does not imply “physicalism true”. Numbers just might not be real entities at all, as in Formalism.
Now suppose that as the game board evolves, the AI discovers Peano Arithmetic, derives “2 + 2 = 4”, and observes that this corresponds to what happens when it puts 2 apples in a bag that already contains 2 apples (there are apple-like things in the AI’s simulation). The fact that the AI derives “2 + 2 = 4″, and the fact that it observes a correspondence between this and the apples, has to be entirely determined by the rules of the Game of Life and the initial state.
If the AI discovers transfinite maths or continuum mechanics, that fact is also entirely determined by rules of the Game of Life and the initial state. And neither of them can apply to a GoL universe—they are not “physics”.
Now, at this point, you need to choose between stipulating that the non-physical maths is false because it is non physical (finitism); or accepting that Platonism and physicalism are both false.
If the AI notices how deep and intricate math is, how its many branches seem to be greatly interconnected with each other, and postulates that math is unreasonably effective
But it’s not maximally effective: maximal effectiveness would mean that any mathematical truth is a physical truth.
You’ll note that it doesn’t matter if we replace the Game of Life by some other similar game where the board is a continuum
If the physical universe is any way a subset of the mathematical “universe” , you have the same problem.
I mean, but our universe is not Conway’s Game of Life.
Setting aside for now the problems with our universe being continuous/quantum weirdness/etc, the bigger issue has to do with the nature of the initial state of the board.
Whether or not math would beunreasonably effective in a universe made out of Conway’s Game of Life depends supremely on the initial state of the board.
If the board was initialized randomly, then it would already be in a maximum-entropy distribution, hence “minds” would have no predictive power and math would not be unreasonably effective. Any minds that did come into existence would be similar to Boltzmann Brains in the sense that they would come into existence for one brief moment and then be destroyed the next.
The initial board would have to be special for minds like ours to exist in Conway’s Game of Life. The initial setup of the board would have to be in a specific configuration that allowed minds to exist for long durations of time and predict things. And in order for that to be the case, there would have to be some universe wide set of rules governing how the board was set up. This is analogous to how the number “2″ is a thing mathematicians think is useful no matter where you go in our universe.
Math isn’t about some local deterministic property that depends on the interaction of simple parts but about the global patterns.
It’s this one.
Given that you’re asking this question, I still haven’t been clear enough. I’ll try to explain it one last time. This time I’ll talk about Conway’s Game of Life and AI. The argument will carry over straightforwardly to physics and humans. (I know that Conway’s Game of life is made up of discrete cells, but I won’t be using that fact in the following argument.)
Suppose there is a Game of Life board which has an initial state which will simulate an AI. Hopefully it is inarguable that the AI’s behavior is entirely determined by the cell states and GoL rules.
Now suppose that as the game board evolves, the AI discovers Peano Arithmetic, derives “2 + 2 = 4”, and observes that this corresponds to what happens when it puts 2 apples in a bag that already contains 2 apples (there are apple-like things in the AI’s simulation). The fact that the AI derives “2 + 2 = 4″, and the fact that it observes a correspondence between this and the apples, has to be entirely determined by the rules of the Game of Life and the initial state.
In case this seems too simple and obvious so far and you’re wondering if you’re missing something, you’re probably not missing anything, this is meant to be simple and obvious.
If the AI notices how deep and intricate math is, how its many branches seem to be greatly interconnected with each other, and postulates that math is unreasonably effective. This also has to be caused entirely by the initial state and rules of the Game of Life. And if the Game of Life board is made up of sets embedded inside some model of set theory, or if it’s not embedded in anything and is just the only thing in all of existence, in either case nothing changes about the AI’s observations or actions and nothing ought to change about its predictions!
And if the existence or non-existence of something changes nothing about what it will observe, then using its existence to “explain” any of its observations is a contradiction in terms. This means that even its observation of the unreasonable effectiveness of math cannot be explained by the existence of a mathematical universe outside of the Game of Life board.
Connecting this back to what I was saying before, the “small parts” here are the cells of the Game of Life. You’ll note that it doesn’t matter if we replace the Game of Life by some other similar game where the board is a continuum. It also doesn’t even matter if the act of translating statements about the AI into statements about the board is uncomputable. All that matters is that the AI’s behavior is entirely determined by the “small parts”.
You might have noticed a loophole in this argument, in that even though the existence of math cannot change anything past the initial board state, if the board was embedded inside a model of set theory, then it would be that model which determined the initial state and rules. However, since the existence of math is compatible with every consistent set of rules and literally every initial board state, knowing this would also give no predictive power to the AI.
At best the AI could try to argue that being embedded inside a mathematical universe explains why the Game of Life rules are consistent. But then it would still be a mystery why the mathematical universe itself follows consistent rules, so in the end the AI would be left with just as many questions as it started with.
Note that “Platonism false” does not imply “physicalism true”. Numbers just might not be real entities at all, as in Formalism.
If the AI discovers transfinite maths or continuum mechanics, that fact is also entirely determined by rules of the Game of Life and the initial state. And neither of them can apply to a GoL universe—they are not “physics”.
Now, at this point, you need to choose between stipulating that the non-physical maths is false because it is non physical (finitism); or accepting that Platonism and physicalism are both false.
But it’s not maximally effective: maximal effectiveness would mean that any mathematical truth is a physical truth.
If the physical universe is any way a subset of the mathematical “universe” , you have the same problem.
I mean, but our universe is not Conway’s Game of Life.
Setting aside for now the problems with our universe being continuous/quantum weirdness/etc, the bigger issue has to do with the nature of the initial state of the board.
Whether or not math would be unreasonably effective in a universe made out of Conway’s Game of Life depends supremely on the initial state of the board.
If the board was initialized randomly, then it would already be in a maximum-entropy distribution, hence “minds” would have no predictive power and math would not be unreasonably effective. Any minds that did come into existence would be similar to Boltzmann Brains in the sense that they would come into existence for one brief moment and then be destroyed the next.
The initial board would have to be special for minds like ours to exist in Conway’s Game of Life. The initial setup of the board would have to be in a specific configuration that allowed minds to exist for long durations of time and predict things. And in order for that to be the case, there would have to be some universe wide set of rules governing how the board was set up. This is analogous to how the number “2″ is a thing mathematicians think is useful no matter where you go in our universe.
Math isn’t about some local deterministic property that depends on the interaction of simple parts but about the global patterns.