Maybe I’m just confused because I recently had an argument with someone who didn’t believe in infinity.
When I pointed out all of physics is based on the assumption that spacetime is continuous (an example of infinity) his response was essentially “we’ll fix that someday”.
So, given that you deny “the mathematical universe”, does that mean you think spacetime isn’t continuous? Or are “small physical parts” allowed to be infinitely subdivided?
Is it really based on an assumption that there’s no quantized layer, even down very deep? I’m not sure I have a belief either way, but I don’t know of any macro-scale implication that requires true continuously-divisible units.
You can’t have e.g. Lorenz Invariance on a quantized spacetime. Of course, you can always argue that “the quantization is really small giving us approximate Lorenz Invariance”, but this would be a claim not only without evidence, but actually against all of the evidence we have.
Similarly, the Schrodinger Wave Equation is defined mathematically as an operator evolving in continuous spacetime. Obviously you can approximate it using a digital world (as we do on our computers), but the real world doesn’t show any evidence of using such an approximation.
My view is compatible with the existence of actual infinities within the physical universe. One potential source of infinity is, as you say, the possibility of infinite subdivision of spacetime. Another is the possibility that spacetime is unboundedly large. I don’t have strong opinions one way or another on if these possibilities are true or not.
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”?
The Unreasonable Effectiveness of Math makes a predictable claim: models which can be represented using concise mathematical notation are more likely to be true, but this includes the whole mathematical universe.
What part of the mathematical universe do you reject exactly?
I’m still trying to understand this quote:
In general I have not found anything which a mathematical universe explained which did not either contradict reductionism, or which could not be explained at least as well using the physical view of the world. I conclude from this that it’s unlikely that there exists a mathematical universe.
And so far it sounds like you’re fine with literal infinity. So what part of a mathematical universe do you find distasteful? Is it all infinities larger than 2ℵ0 or the idea that “2” exists as an abstract idea apart from any physical model, or something else?
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.
Maybe I’m just confused because I recently had an argument with someone who didn’t believe in infinity.
When I pointed out all of physics is based on the assumption that spacetime is continuous (an example of infinity) his response was essentially “we’ll fix that someday”.
So, given that you deny “the mathematical universe”, does that mean you think spacetime isn’t continuous? Or are “small physical parts” allowed to be infinitely subdivided?
Is it really based on an assumption that there’s no quantized layer, even down very deep? I’m not sure I have a belief either way, but I don’t know of any macro-scale implication that requires true continuously-divisible units.
You can’t have e.g. Lorenz Invariance on a quantized spacetime. Of course, you can always argue that “the quantization is really small giving us approximate Lorenz Invariance”, but this would be a claim not only without evidence, but actually against all of the evidence we have.
Similarly, the Schrodinger Wave Equation is defined mathematically as an operator evolving in continuous spacetime. Obviously you can approximate it using a digital world (as we do on our computers), but the real world doesn’t show any evidence of using such an approximation.
My view is compatible with the existence of actual infinities within the physical universe. One potential source of infinity is, as you say, the possibility of infinite subdivision of spacetime. Another is the possibility that spacetime is unboundedly large. I don’t have strong opinions one way or another on if these possibilities are true or not.
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”?
The Unreasonable Effectiveness of Math makes a predictable claim: models which can be represented using concise mathematical notation are more likely to be true, but this includes the whole mathematical universe.
What part of the mathematical universe do you reject exactly?
I’m still trying to understand this quote:
And so far it sounds like you’re fine with literal infinity. So what part of a mathematical universe do you find distasteful? Is it all infinities larger than 2ℵ0 or the idea that “2” exists as an abstract idea apart from any physical model, or something else?
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.