The only way in which we can access an abstract mathematical fact, learn about its truth, is through some physical tools, such that we believe their behavior to be linked to the abstract statement.
In this case, we believe that if 5324 + 2326 = 7650, then adding 5324 apples and 2326 apples gives 7650 apples; and if 5324 + 2326 = 9650, then adding 5324 apples and 2326 apples gives 9650 apples; and if adding 5324 apples and 2326 apples gives 7650 apples, then 5324 + 2326 = 7650.
If instead of normal apples, we have magical apples that don’t add, then we probably wouldn’t believe that this particular abstract statement is related to the apples, and that observing the apples tells us something about the statement. On the other hand, we’d still need some tool to access the abstract fact in order to reason about it, perhaps a brain or weird-physics-based calculator if no simple objects comply. If such is not available, then we can’t answer (or even ask) the question.
The only way in which we can access an abstract mathematical fact, learn about its truth, is through some physical tools, such that we believe their behavior to be linked to the abstract statement.
This has not been my experience in mathematics. There’s lots and lots of theorems that I know where I don’t have any physical counterpart for the claim. Cantor’s diagonalization argument, for instance. The irrationality of the square root of two. The fundamental theorem of arithmetic, aka the unique prime factorization theorem.
For natural-number arithmetic, geometry, and calculus, it’s often easy to find close physical analogs. For the rest of math, I don’t see it.
Are you using “physical tools” or “linked to” in a much broader sense than I understand?
This has not been my experience in mathematics. There’s lots and lots of theorems that I know where I don’t have any physical counterpart for the claim.
You have your physical brain (I did mention this option in the comment). The “counterpart” can follow a syntactic description of the semantic fact, for example, which is how the axiomatic method normally works (but there are other options, the important thing is that you (correctly) believe the tool to be controlled by the abstract fact, as you believe your mathematical reasoning to be controlled by abstract facts).
There was a thought related to this that I’ve been trying to untangle. Assuming the brain is computable, which seems to me to be quite a safe assumption, doesn’t that imply that all of mathematics should be able to be framed computationally? It seems obvious to me that there should be some algorithm that could create mathematics by organizing a stream of input data in some fashion. After all, there is a (presumably) computable subsystem of Earth that does that, namely the mathematics community.
That said, in what light are we to consider “incomputable” mathematics? It seems like this is drawing a meaningless distinction given the computable universe hypothesis. Perhaps it can be thought of as a sort of counterfactual reasoning? This certainly seems to be the case with, say, accelerated Turing machines, the counterfactual being “what if there were no universal speed cap?”
Apologies if I’m not making sense, I’ve been awake for quite a while.
The position that makes the most sense to me is that uncomputable mathematics exist in a platonic sense, we just happen to inhabit a part of the ultimate ensemble that appears computable, but (1) there’s a chance that our universe is actually uncomputable (or alternatively, we’re “simultaneously” in computable and uncomputable parts of the multiverse), and (2) we can use computable machinery to learn some facts about uncomputable math.
The alternative of saying that only computable math exists seems to imply that all of the work done on understanding uncomputable math (such as Turing degrees and the like) is merely a game of manipulating meaningless symbols. In other words, those mathematicians who think they are reasoning about uncomputable math are not actually doing anything meaningful. That is hard to swallow.
What do you mean by computable or uncomputable math? In one sense, all math is computable (you can enumerate all valid proofs in a formal system, even if it talks about something mind-bogglingly huge like Grothendieck universes). In another sense, all math is uncomputable (you can’t enumerate all true facts about the integers, never mind more complex stuff). Is there some well-defined third way of cashing out the concept?
I don’t have an answer for that, since I’m not proposing to make a distinction based on computable vs uncomputable math. But I think Juergen Schmidhuber has defended some version of “only computable math exists” on the everything-list, so you can try searching the archives there.
It seems like your conception is isomorphic to mine to the extent that the ultimate ensemble can be considered as a set of counterfactual (with respect to local physical law) universes. You’ll see below that I brought up Zeno machines, and noted that they seem to follow this paradigm. Does this seem accurate?
The alternative of saying that only computable math exists seems to imply that all of the work done on understanding uncomputable math (such as Turing degrees and the like) is merely a game of manipulating meaningless symbols. In other words, those mathematicians who think they are reasoning about uncomputable math are not actually doing anything meaningful. That is hard to swallow.
Even the most abstract symbol manipulation can turn out to be functionally equivalent to a more concrete mode of reasoning and hence be quite useful. It is still an exploration of some sort of algorithmic reasoning, and coming from the view that mathematics is something like the art of predicting the outcome of various manipulations of structured data, such abstract games are not totally invalid.
My intuition is currently along the lines of thinking that uncomputable mathematics is disguised computable math, since the manipulation and creation of the system itself is computable (again, given the assumption that the human mind is computable), and so it is perfectly valid but misinterpreted in some sense.
I feel that I should say that my primary motivation for this line of thought is a desire to think about how a machine might create mathematics from scratch. I’ve started reading Simon Colton’s work, it seems quite interesting. Perhaps you could offer some insight into this sort of issue?
Talking or reasoning about the uncomputable isn’t the same as “computing” the uncomputable. The first may very well be computable while the second obviously isn’t.
Talking or reasoning about the uncomputable isn’t the same as “computing” the uncomputable.
Obviously, but I didn’t mean to imply that it was. My question is what we are actually reasoning about when we are reasoning about the uncomputable. Apologies if this wasn’t clear.
It seems to me that oracle computations and accelerated Turing machines, seem to be related to counterfactual reasoning in the sense that they suppose things that are not the case such as Galilean velocity addition or that we can obtain the result of a non-halting computation in finite time and use that to compute further results.
You may have heard of Chaitin’s constant, an uncomputable number. It’s easy to define the number, but it’s impossible for machines or people to compute it: For every person, there is a number N such that that person will never be able to figure out what the Nth digit of Chaitin’s constant is. I hope this helps.
Well, as I understand it Chaitin numbers are computable with a Zeno machine or an oracle machine with a halting oracle as they are Delta_20 in the arithmetic hierarchy. This still seems to follow my previous line of thinking; that reasoning about incomputable mathematics is itself computable (of course), but (more interestingly) can be framed as reasoning with counterfactuals about physical law (what if I could break the speed of light, in this case).
I would also note that the reals would be countable if it weren’t for incomputable numbers. Any number that can be approximated to an arbitrary degree by an algorithm can of course be represented by that algorithm, and thus the set of all such numbers is equivalent to a subset of Turing machines, which are countable.
All this considered, I’m not really sure how Chaitin’s constant (at least, in particular, beyond the fact that it is incomputable) bears on the issue.
Since this notion of our brains being calculators of—and thereby providing evidence about—certain abstract mathematical facts seems transplanted from Eliezer’s metaethics, I wonder if there are any important differences between these two ideas, i.e. between trying to answer “5324 + 2326 = ?” and “is X really the right thing to do?”.
In other words, are moral questions just a subset of math living in a particularly complex formal system (our “moral frame of reference”) or are they a beast of a different-but-similar-enough-to-become-confused kind? Is the apparent similarity just a consequence of both “using” the same (metaphorical) math/truth-discovering brain module?
“Controlled by abstract fact” as in Controlling Constant Programs idea?
That post gives a toy model, but the objects of acausal control can be more general: they are not necessarily natural numbers (computed by some program), they can be other kinds of mathematical structures; their definitions don’t have to come in the form of programs that compute them, or syntactic axioms; we might have no formal definitions that are themselves understood, so that we can only use the definitions, without knowing how they work; and we might even have no ability to refer to the exact object of control, and instead work with an approximation.
Also, consider that even when you talk of “explicit” tools for accessing abstract facts, such as apples or calculators or theorems written of a sheet of paper, these tools are still immensely complicated physical objects, and what you mean by saying that they are simple is that you understand how they are related to simple abstract descriptions, which you understand. So the only difference between understanding what a “real number” is “directly”, and pointing to an axiomatic definition, is that you are comfortable with intuitive understanding of “axiomatic definition”, but it too is an abstract idea that you could further describe using another level of axiomatic definition. And in fact considering syntactic tools as mathematical structures led to generalizing from finite logical statements to infinite ones, see infinitary logic.
This kind of thinking (that I actually quite like) presupposes the existence of abstract mathematical facts, but I’m at loss figuring out in what part of the territory are they stored, and if this is a wrong question to ask, what precisely does it make it so.
Does postulating their existence buy us anything else than TDT’s nice decision procedure?
The only way in which we can access an abstract mathematical fact, learn about its truth, is through some physical tools, such that we believe their behavior to be linked to the abstract statement.
In this case, we believe that if 5324 + 2326 = 7650, then adding 5324 apples and 2326 apples gives 7650 apples; and if 5324 + 2326 = 9650, then adding 5324 apples and 2326 apples gives 9650 apples; and if adding 5324 apples and 2326 apples gives 7650 apples, then 5324 + 2326 = 7650.
If instead of normal apples, we have magical apples that don’t add, then we probably wouldn’t believe that this particular abstract statement is related to the apples, and that observing the apples tells us something about the statement. On the other hand, we’d still need some tool to access the abstract fact in order to reason about it, perhaps a brain or weird-physics-based calculator if no simple objects comply. If such is not available, then we can’t answer (or even ask) the question.
This has not been my experience in mathematics. There’s lots and lots of theorems that I know where I don’t have any physical counterpart for the claim. Cantor’s diagonalization argument, for instance. The irrationality of the square root of two. The fundamental theorem of arithmetic, aka the unique prime factorization theorem.
For natural-number arithmetic, geometry, and calculus, it’s often easy to find close physical analogs. For the rest of math, I don’t see it.
Are you using “physical tools” or “linked to” in a much broader sense than I understand?
You have your physical brain (I did mention this option in the comment). The “counterpart” can follow a syntactic description of the semantic fact, for example, which is how the axiomatic method normally works (but there are other options, the important thing is that you (correctly) believe the tool to be controlled by the abstract fact, as you believe your mathematical reasoning to be controlled by abstract facts).
There was a thought related to this that I’ve been trying to untangle. Assuming the brain is computable, which seems to me to be quite a safe assumption, doesn’t that imply that all of mathematics should be able to be framed computationally? It seems obvious to me that there should be some algorithm that could create mathematics by organizing a stream of input data in some fashion. After all, there is a (presumably) computable subsystem of Earth that does that, namely the mathematics community.
That said, in what light are we to consider “incomputable” mathematics? It seems like this is drawing a meaningless distinction given the computable universe hypothesis. Perhaps it can be thought of as a sort of counterfactual reasoning? This certainly seems to be the case with, say, accelerated Turing machines, the counterfactual being “what if there were no universal speed cap?”
Apologies if I’m not making sense, I’ve been awake for quite a while.
The position that makes the most sense to me is that uncomputable mathematics exist in a platonic sense, we just happen to inhabit a part of the ultimate ensemble that appears computable, but (1) there’s a chance that our universe is actually uncomputable (or alternatively, we’re “simultaneously” in computable and uncomputable parts of the multiverse), and (2) we can use computable machinery to learn some facts about uncomputable math.
The alternative of saying that only computable math exists seems to imply that all of the work done on understanding uncomputable math (such as Turing degrees and the like) is merely a game of manipulating meaningless symbols. In other words, those mathematicians who think they are reasoning about uncomputable math are not actually doing anything meaningful. That is hard to swallow.
What do you mean by computable or uncomputable math? In one sense, all math is computable (you can enumerate all valid proofs in a formal system, even if it talks about something mind-bogglingly huge like Grothendieck universes). In another sense, all math is uncomputable (you can’t enumerate all true facts about the integers, never mind more complex stuff). Is there some well-defined third way of cashing out the concept?
I don’t have an answer for that, since I’m not proposing to make a distinction based on computable vs uncomputable math. But I think Juergen Schmidhuber has defended some version of “only computable math exists” on the everything-list, so you can try searching the archives there.
It seems like your conception is isomorphic to mine to the extent that the ultimate ensemble can be considered as a set of counterfactual (with respect to local physical law) universes. You’ll see below that I brought up Zeno machines, and noted that they seem to follow this paradigm. Does this seem accurate?
Even the most abstract symbol manipulation can turn out to be functionally equivalent to a more concrete mode of reasoning and hence be quite useful. It is still an exploration of some sort of algorithmic reasoning, and coming from the view that mathematics is something like the art of predicting the outcome of various manipulations of structured data, such abstract games are not totally invalid.
My intuition is currently along the lines of thinking that uncomputable mathematics is disguised computable math, since the manipulation and creation of the system itself is computable (again, given the assumption that the human mind is computable), and so it is perfectly valid but misinterpreted in some sense.
I feel that I should say that my primary motivation for this line of thought is a desire to think about how a machine might create mathematics from scratch. I’ve started reading Simon Colton’s work, it seems quite interesting. Perhaps you could offer some insight into this sort of issue?
Talking or reasoning about the uncomputable isn’t the same as “computing” the uncomputable. The first may very well be computable while the second obviously isn’t.
Obviously, but I didn’t mean to imply that it was. My question is what we are actually reasoning about when we are reasoning about the uncomputable. Apologies if this wasn’t clear.
It seems to me that oracle computations and accelerated Turing machines, seem to be related to counterfactual reasoning in the sense that they suppose things that are not the case such as Galilean velocity addition or that we can obtain the result of a non-halting computation in finite time and use that to compute further results.
You may have heard of Chaitin’s constant, an uncomputable number. It’s easy to define the number, but it’s impossible for machines or people to compute it: For every person, there is a number N such that that person will never be able to figure out what the Nth digit of Chaitin’s constant is. I hope this helps.
Well, as I understand it Chaitin numbers are computable with a Zeno machine or an oracle machine with a halting oracle as they are Delta_20 in the arithmetic hierarchy. This still seems to follow my previous line of thinking; that reasoning about incomputable mathematics is itself computable (of course), but (more interestingly) can be framed as reasoning with counterfactuals about physical law (what if I could break the speed of light, in this case).
I would also note that the reals would be countable if it weren’t for incomputable numbers. Any number that can be approximated to an arbitrary degree by an algorithm can of course be represented by that algorithm, and thus the set of all such numbers is equivalent to a subset of Turing machines, which are countable.
All this considered, I’m not really sure how Chaitin’s constant (at least, in particular, beyond the fact that it is incomputable) bears on the issue.
I think I was confused about what you were confused about.
“Controlled by abstract fact” as in Controlling Constant Programs idea?
Since this notion of our brains being calculators of—and thereby providing evidence about—certain abstract mathematical facts seems transplanted from Eliezer’s metaethics, I wonder if there are any important differences between these two ideas, i.e. between trying to answer “5324 + 2326 = ?” and “is X really the right thing to do?”.
In other words, are moral questions just a subset of math living in a particularly complex formal system (our “moral frame of reference”) or are they a beast of a different-but-similar-enough-to-become-confused kind? Is the apparent similarity just a consequence of both “using” the same (metaphorical) math/truth-discovering brain module?
That post gives a toy model, but the objects of acausal control can be more general: they are not necessarily natural numbers (computed by some program), they can be other kinds of mathematical structures; their definitions don’t have to come in the form of programs that compute them, or syntactic axioms; we might have no formal definitions that are themselves understood, so that we can only use the definitions, without knowing how they work; and we might even have no ability to refer to the exact object of control, and instead work with an approximation.
Also, consider that even when you talk of “explicit” tools for accessing abstract facts, such as apples or calculators or theorems written of a sheet of paper, these tools are still immensely complicated physical objects, and what you mean by saying that they are simple is that you understand how they are related to simple abstract descriptions, which you understand. So the only difference between understanding what a “real number” is “directly”, and pointing to an axiomatic definition, is that you are comfortable with intuitive understanding of “axiomatic definition”, but it too is an abstract idea that you could further describe using another level of axiomatic definition. And in fact considering syntactic tools as mathematical structures led to generalizing from finite logical statements to infinite ones, see infinitary logic.
This kind of thinking (that I actually quite like) presupposes the existence of abstract mathematical facts, but I’m at loss figuring out in what part of the territory are they stored, and if this is a wrong question to ask, what precisely does it make it so.
Does postulating their existence buy us anything else than TDT’s nice decision procedure?
I can infer stuff about mathematical facts, and some of the facts are morally relevant. Whether they “exist” is of unclear relevance.