I don’t know that there is an official “LessWrongist” philosophy for math.
In some posts, Yudkowsky reads to me as being a Platonist/realist. In other posts, his philosophy comes across to me as some kind of Intuitionism or Fictionalism. I don’t recall reading anything where it is clearly stated.
Modal Structuralism
In my own study of this, guided by another rationalist who had conducted an extensive study of the philosophy of math, I concluded that modal structuralism is correct. This view is mostly associated with the philosopher Geoffrey Hellman.
Structuralism is a group of philosophies of math that hold that mathematical objects are exhaustively defined by their place in mathematical structures. For example, the number 2 doesn’t exist on its own and have a property of “twoness”, rather it is defined by being in the second position in the structure of natural numbers. Its only property is in its place and how that place relates to the rest of the structure. Structuralism is an epistemologically realistic philosophy. So it holds that the hash of a busy beaver number example you gave does have a truth value, even if we can never determine it. However, structuralism doesn’t say what kind of existence mathematical structures have. So there are subvarieties for the different choices there. The book “Mathematical Structuralism” by Geoffrey Helman and Stewart Shapiro is a good introduction to structuralism.
Modal structuralism holds that mathematical structures do not exist as abstract entities (e.g., in a platonic sense). Instead that, if they did exist, then they would have certain properties. So when we say that your problem of the hash of a busy beaver number has a truth value, we are saying, if the mathematical structures necessary to talk about that problem existed, then the statement would be true or false and further that it would have the same truth value in any possible embodiment since the truth is determined by the properties of the structure. When we say a mathematical structure “exists”, we are really saying that it is logically possible it could exist and that we are talking about the properties it would have if it did exist.
Other Questions
Modal structuralism doesn’t specifically address your other questions about where mathematics comes from, how we evolved humans can practice it, and the unreasonable effectiveness of mathematics. However, I think there are some reasonable answers that are consistent with it. I’ll try to sketch them out.
Math is the study of the properties of mathematical structures if they were to exist. One way to define these structures is via systems of rules (i.e. axioms). Since the universe follows logically consistent rules (i.e. physical laws), it will be possible to, at least partially, map mathematical structures onto the physical systems. (We chose to study the ones that correspond to our universe first since those were the ones of most interest to us.) As to why the universe has logically consistent laws rather than chaos, I don’t have a good answer. I would guess that it is an anthropic argument. It isn’t possible to have thinking beings in a universe without logically consistent laws.
Since the universe follows logically consistent laws and we evolved to thrive in that universe, we evolved skills like logic, counting, etc. These were useful to our survival. There seems to have been some kind of runaway intelligence competition (sexual or social selection?). That intelligence built on the foundation of logic gave us the skills we need to do mathematics. Since it is modal reasoning about what would be true of mathematical structures if they existed, we don’t need any magic ability to know or get in touch with mathematical objects. Just our logic, counting, etc. skills are enough for us to make statements about math.
Does structuralism hold that the statement of the Continuum Hypothesis has a truth value? If no, how does it differentiate between my hash of BB mod 2 statement and CH?
I think there may be some variation in the answer across the different strands of structuralism.
Modal structuralism would say that there are two different possible mathematical structures. One where the CH is true and one where the negation of CH is true. Any statements dependent on the CH will have to be qualified with the mathematical structure being studied. If there is some statement X that is true if CH but false if not CH, then that is fine. It is no different than saying the angles of a triangle add to 180 in Euclidean space but not in hyperbolic space.
Your hash of BB mod 2 statement is different because it is true in all the mathematical structures we study (or not expressible in certain restricted structures). Also, it is not independent of those structures. So we simply say it is true. But if there were some logically possible mathematical structure where it had the opposite truth value, then one would need to qualify which mathematical structure one was referring to when talking about its truth.
I don’t know that there is an official “LessWrongist” philosophy for math.
In some posts, Yudkowsky reads to me as being a Platonist/realist. In other posts, his philosophy comes across to me as some kind of Intuitionism or Fictionalism. I don’t recall reading anything where it is clearly stated.
Modal Structuralism
In my own study of this, guided by another rationalist who had conducted an extensive study of the philosophy of math, I concluded that modal structuralism is correct. This view is mostly associated with the philosopher Geoffrey Hellman.
Structuralism is a group of philosophies of math that hold that mathematical objects are exhaustively defined by their place in mathematical structures. For example, the number 2 doesn’t exist on its own and have a property of “twoness”, rather it is defined by being in the second position in the structure of natural numbers. Its only property is in its place and how that place relates to the rest of the structure. Structuralism is an epistemologically realistic philosophy. So it holds that the hash of a busy beaver number example you gave does have a truth value, even if we can never determine it. However, structuralism doesn’t say what kind of existence mathematical structures have. So there are subvarieties for the different choices there. The book “Mathematical Structuralism” by Geoffrey Helman and Stewart Shapiro is a good introduction to structuralism.
Modal structuralism holds that mathematical structures do not exist as abstract entities (e.g., in a platonic sense). Instead that, if they did exist, then they would have certain properties. So when we say that your problem of the hash of a busy beaver number has a truth value, we are saying, if the mathematical structures necessary to talk about that problem existed, then the statement would be true or false and further that it would have the same truth value in any possible embodiment since the truth is determined by the properties of the structure. When we say a mathematical structure “exists”, we are really saying that it is logically possible it could exist and that we are talking about the properties it would have if it did exist.
Other Questions
Modal structuralism doesn’t specifically address your other questions about where mathematics comes from, how we evolved humans can practice it, and the unreasonable effectiveness of mathematics. However, I think there are some reasonable answers that are consistent with it. I’ll try to sketch them out.
Math is the study of the properties of mathematical structures if they were to exist. One way to define these structures is via systems of rules (i.e. axioms). Since the universe follows logically consistent rules (i.e. physical laws), it will be possible to, at least partially, map mathematical structures onto the physical systems. (We chose to study the ones that correspond to our universe first since those were the ones of most interest to us.) As to why the universe has logically consistent laws rather than chaos, I don’t have a good answer. I would guess that it is an anthropic argument. It isn’t possible to have thinking beings in a universe without logically consistent laws.
Since the universe follows logically consistent laws and we evolved to thrive in that universe, we evolved skills like logic, counting, etc. These were useful to our survival. There seems to have been some kind of runaway intelligence competition (sexual or social selection?). That intelligence built on the foundation of logic gave us the skills we need to do mathematics. Since it is modal reasoning about what would be true of mathematical structures if they existed, we don’t need any magic ability to know or get in touch with mathematical objects. Just our logic, counting, etc. skills are enough for us to make statements about math.
Does structuralism hold that the statement of the Continuum Hypothesis has a truth value? If no, how does it differentiate between my hash of BB mod 2 statement and CH?
I think there may be some variation in the answer across the different strands of structuralism.
Modal structuralism would say that there are two different possible mathematical structures. One where the CH is true and one where the negation of CH is true. Any statements dependent on the CH will have to be qualified with the mathematical structure being studied. If there is some statement X that is true if CH but false if not CH, then that is fine. It is no different than saying the angles of a triangle add to 180 in Euclidean space but not in hyperbolic space.
Your hash of BB mod 2 statement is different because it is true in all the mathematical structures we study (or not expressible in certain restricted structures). Also, it is not independent of those structures. So we simply say it is true. But if there were some logically possible mathematical structure where it had the opposite truth value, then one would need to qualify which mathematical structure one was referring to when talking about its truth.
I would suspect LessWrongers would tend to explain it by Solomonoff induction