For all mathematical theorems can be restated in the form:
If the axioms A, B, and C and the conditions X, Y and Z are satisfied, then the statement Q is also true.
Therefore, in any situations where the statements A,B,C and X,Y,Z are true, you will expect Q to also be verified.
In other words, mathematical statements automatically pay rent in terms of changing what you expect. (Which is) the very thing it was required to show. ■
In practice:
If you demonstrate Pythagoras’s Theorem, and you calculate that 3^2+4^2=5^2, you will expect a certain method of getting right angles to work.
If you exhibit the aperiodic Penrose Tiling, you will expect Quasicrystals to exist.
If you demonstrate the impossibility of solving to the Halting Problem, you will not expect even a hypothetical hyperintelligence to be able to solve it.
If you understand why you can’t trisect an angle with an unmarked ruler and a compass (not both used at the same time), you will know immediately that certain proofs are going to be wrong.
and so on and so forth.
Yes, we might not immediately know where a given mathematical fact will come in handy when observing the world, but by their nature, mathematical facts tell us exactly when to expect them.
Is this to say that one of the purposes of mathematics is to prove something new, even without knowing what it might be used for, with the awareness that it might be useful at a later point? Or that it might form part of a proof for something else that is also currently unknown?
Yes, there are numerous cases where a field in “pure” mathematics proved interesting theorems that mathematicians undertook because of its challenging and elegant nature (like certain theorems possess generality and elegance) which were then to be found to be practically useful, which are called “applied” mathematics. Frankly, this distinction is blurred as pure mathematics are so useful (see Eugene Wigner’s “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”) that the abstract nature of mathematics has huge extensibility and general applications in multiple domains. For instance, Einstein’s GR was based on the pure mathematics of Riemannian manifolds, which is an abstract topological structure, not tied to reality in any way initially. Or how algebraic topology is used for data mining, how number theory is used for cryptography, how linear algebra is used for machine learning, group theory is used for particle physics… and even how Bayesian probability theory is used for LW rationality.
Stephen Wolfram has great resources on rulial spaces and the nature of computation for the universe’s fundamental ontology (the territory not the map) in which these networks of theorems can correspond to our empirical reality. (psst I am a very new LW user, and I am deciding if I should do a Sequence for this idea of “rulial cover” which is how rulial deduction can be applied to Solomonoff induction and Bayesian abduction, would be great if someone thinks this is interesting to explore so I can be motivated)
To link back to Eliezer’s post, “floating beliefs” in a Bayesian net can be connected through adjusting the “weights” of the edges that connects that belief using Bayesian inference, and mathematics make these robust inferences from axioms (deductively validity as 100% in weight and 0% in prior). Therefore, anticipation becomes certain under a set of idealized axioms.
‘I am deciding if I should do a Sequence for this idea of “rulial cover” which is how rulial deduction can be applied to Solomonoff induction and Bayesian abduction’
I don’t really know what you mean, but if it’s something unseen you can expect it to be useful!
Is it not the purpose of math to tell us “how” to connect things? At the bottom, there are some axioms that we accept as basis of the model, and using another formal model we can infer what to expect from anything whose behavior matches our axioms.
Math makes it very hard to reason about models incorrectly. That’s why it’s good.
Even parts of math that seem particularly outlandish and disconnected just build a higher-level framework on top of more basic concepts that have been successfully utilized over and over again.
That gives us a solid framework on which we can base our reasoning about abstract ideas. Just a few decades ago most people believed the theory of probability was just a useless mathematical game, disconnected from any empirical reality. Now people like you and me use it every day to quantify uncertainty and make better decisions. The connections are not always obvious.
Thats exactly how i felt in high school. Im glad i changed that because it wouldn’t be useful to me if i’d never learned algebra. The first part of the class is hard to use and discouraging to new students.
IMO the distinction between pure and applied math is artificial, or at least contingent; today’s pure math may be tomorrow’s applied math. This point was made in VKS’s comment referenced above:
Yes, we might not immediately know where a given mathematical fact will come in handy when observing the world, but by their nature, mathematical facts tell us exactly when to expect them
The question is whether anyone should believe pure maths now. If you are allowed to believe things that might possibly pay off, then the criterion excludes nothing.
Metabeleifs! Applied math concepts that seem useless now, have, in the past, become useful. Therefore, the belief that “believing in applied math concepts pays rent in experience” pays rent in experience, so therefore you should believe it.
If you believe in applied math, what are the grounds for excluding “pure” math? Most of the time “pure” just means that the mathematician makes no explicit reference to real-world applications and that the theorems are formulated in an abstract setting. Abstraction usually just boils down to figuring out exactly which hypotheses are necessary to get the conclusion you want and then dispensing with the rest.
Let’s take the theory of probability as an example. There’s nothing in the general theory that contradicts everyday, real-world probability applications. Most of the time the general theory does little other than make precise our intuitive notions and avoid the paradoxes that plague a naive approach. This is an artifact of our insistence on logic. A thorough, logical examination of just about any piece of mathematics will quickly lead to the domain “pure” math.
There are beliefs that directly pay rent, and then there are beliefs that are logical consequences of rent-paying beliefs. The same basic principles that give you applied math will also lead to pure math. We can justify spending effort on pure math on the grounds that it may pay off in the future. However, our belief in pure math is tied to our belief in logic.
If you asked whether this can be applied to something like astrology, I’d ask whether astrology was a logical consequence of beliefs that do pay rent.
Unlike scientific knowledge or other beliefs about the material world, a mathematical fact (e.g. that z follows from X1, X2,..., Xn), once proven, is beyond dispute; there is no chance that such a fact will be contradicted by future observations. One is allowed to believe mathematical facts (once proven) because they are indisputably true; that these facts pay rent is supported by VKS’s argument.
Truths of pure maths don’t pay rent in terms iof expected experience. EY has put forward a criterion of truth, correspondence, and a criterion of believability, expected experience , and pure maths fits neither. He didn’t want that to happen, and the problem remains, here and elsewhere, of how to include abstract maths and still exclude the things you don’t like. This is old ground, that the logical postivists went over in the mid 20th century.
My initial interpretation of EY’s original post is that he was explicating a scientific standard of belief that would make sense in many situations, including in reasoning about the physical world (EY’s initial examples were physical phenomena—trees falling, bowling balls dropping, phlogiston, etc.). I did not really think he was proposing the only standard of belief. This is why I was baffled by your insistence that unless a mathematical fact had made successful predictions about physical, observable phenomena, it should be evicted.
However, later in the original post EY used an example out of literary criticism, and here he appears to be applying the standard to mathematics. So, you may be on to something—perhaps EY did intend the standard to be universally applied.
It seems to me that applying EY’s standard too broadly is tantamount to scientism (which I suspect is more-less the point you were making).
Of course astrological claims pay rent. The problem with astrology is not that it’s meaningless but that it’s false, and the problem with astrologers is that they don’t pay the epistemological rent.
Also, a proof is a different thing from a mathematician saying so. The rent that is being paid there is not merely that the theorem will be asserted but that there will be a proof.
The original post does not mention astrology. If you want to spy out some place where Eliezer has said that astrological claims are meaningless, go right ahead. I am not particularly concerned with whether he has or not.
Here and now, you are talking to me, and as I pointed out, the belief can pay rent, but astrologers are not making it do so. Those who have seriously looked for evidence, have, so I understand, generally found the beliefs false.
I think this is both right and not in contradiction with the post.
The belief that pays the rent here is that there is going to be a high correlation between Mars being in conjunction with Jupiter and astrology believers born around August experiencing heightened feelings of being in danger.
That does not say anything on the “truth” of astrology itself.
Same applies to the article’s example on Wulky Wilkinsen. The belief that alienated resublimation justifies the fictional author’s retropositionality does not pay rent. The belief that failing to mention retropositionality correlates with higher chances of failing a literature test on Wilkinsen does probably pay rent.
That was the point. Its a cheat to expect astrology truths to product experiences of reading written materials about astrology, so it’s a cheat expect to pure maths truths …
That was the point. Its a cheat to expect astrology truths to product experiences of reading written materials about astrology, so it’s a cheat expect to pure maths truths …
Let me complete the ellipsis with what I actually said. A mathematical assertion leads me to expect a proof. Not merely experiences of reading written materials repeating the assertion.
What good is math if people don’t know what to connect it to?
All math pays rent.
For all mathematical theorems can be restated in the form:
If the axioms A, B, and C and the conditions X, Y and Z are satisfied, then the statement Q is also true.
Therefore, in any situations where the statements A,B,C and X,Y,Z are true, you will expect Q to also be verified.
In other words, mathematical statements automatically pay rent in terms of changing what you expect. (Which is) the very thing it was required to show. ■
In practice:
If you demonstrate Pythagoras’s Theorem, and you calculate that 3^2+4^2=5^2, you will expect a certain method of getting right angles to work.
If you exhibit the aperiodic Penrose Tiling, you will expect Quasicrystals to exist.
If you demonstrate the impossibility of solving to the Halting Problem, you will not expect even a hypothetical hyperintelligence to be able to solve it.
If you understand why you can’t trisect an angle with an unmarked ruler and a compass (not both used at the same time), you will know immediately that certain proofs are going to be wrong.
and so on and so forth.
Yes, we might not immediately know where a given mathematical fact will come in handy when observing the world, but by their nature, mathematical facts tell us exactly when to expect them.
Is this to say that one of the purposes of mathematics is to prove something new, even without knowing what it might be used for, with the awareness that it might be useful at a later point? Or that it might form part of a proof for something else that is also currently unknown?
Yes, there are numerous cases where a field in “pure” mathematics proved interesting theorems that mathematicians undertook because of its challenging and elegant nature (like certain theorems possess generality and elegance) which were then to be found to be practically useful, which are called “applied” mathematics. Frankly, this distinction is blurred as pure mathematics are so useful (see Eugene Wigner’s “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”) that the abstract nature of mathematics has huge extensibility and general applications in multiple domains. For instance, Einstein’s GR was based on the pure mathematics of Riemannian manifolds, which is an abstract topological structure, not tied to reality in any way initially. Or how algebraic topology is used for data mining, how number theory is used for cryptography, how linear algebra is used for machine learning, group theory is used for particle physics… and even how Bayesian probability theory is used for LW rationality.
Stephen Wolfram has great resources on rulial spaces and the nature of computation for the universe’s fundamental ontology (the territory not the map) in which these networks of theorems can correspond to our empirical reality. (psst I am a very new LW user, and I am deciding if I should do a Sequence for this idea of “rulial cover” which is how rulial deduction can be applied to Solomonoff induction and Bayesian abduction, would be great if someone thinks this is interesting to explore so I can be motivated)
To link back to Eliezer’s post, “floating beliefs” in a Bayesian net can be connected through adjusting the “weights” of the edges that connects that belief using Bayesian inference, and mathematics make these robust inferences from axioms (deductively validity as 100% in weight and 0% in prior). Therefore, anticipation becomes certain under a set of idealized axioms.
‘I am deciding if I should do a Sequence for this idea of “rulial cover” which is how rulial deduction can be applied to Solomonoff induction and Bayesian abduction’
I don’t really know what you mean, but if it’s something unseen you can expect it to be useful!
Is it not the purpose of math to tell us “how” to connect things? At the bottom, there are some axioms that we accept as basis of the model, and using another formal model we can infer what to expect from anything whose behavior matches our axioms.
Math makes it very hard to reason about models incorrectly. That’s why it’s good. Even parts of math that seem particularly outlandish and disconnected just build a higher-level framework on top of more basic concepts that have been successfully utilized over and over again.
That gives us a solid framework on which we can base our reasoning about abstract ideas. Just a few decades ago most people believed the theory of probability was just a useless mathematical game, disconnected from any empirical reality. Now people like you and me use it every day to quantify uncertainty and make better decisions. The connections are not always obvious.
http://abstrusegoose.com/504 :-)
Thats exactly how i felt in high school. Im glad i changed that because it wouldn’t be useful to me if i’d never learned algebra. The first part of the class is hard to use and discouraging to new students.
Is pure math a set of beliefs that should be evicted?
No, for reasons expressed above by VKS.
Note the word “pure”. By definition, pure maths doesn’t pay off in experience. If it did, it would be applied.
IMO the distinction between pure and applied math is artificial, or at least contingent; today’s pure math may be tomorrow’s applied math. This point was made in VKS’s comment referenced above:
The question is whether anyone should believe pure maths now. If you are allowed to believe things that might possibly pay off, then the criterion excludes nothing.
Metabeleifs! Applied math concepts that seem useless now, have, in the past, become useful. Therefore, the belief that “believing in applied math concepts pays rent in experience” pays rent in experience, so therefore you should believe it.
If you believe in applied math, what are the grounds for excluding “pure” math? Most of the time “pure” just means that the mathematician makes no explicit reference to real-world applications and that the theorems are formulated in an abstract setting. Abstraction usually just boils down to figuring out exactly which hypotheses are necessary to get the conclusion you want and then dispensing with the rest.
Let’s take the theory of probability as an example. There’s nothing in the general theory that contradicts everyday, real-world probability applications. Most of the time the general theory does little other than make precise our intuitive notions and avoid the paradoxes that plague a naive approach. This is an artifact of our insistence on logic. A thorough, logical examination of just about any piece of mathematics will quickly lead to the domain “pure” math.
I am not making the statement “exclude pure math”, I am posing the question “if pure math stays, what else stays?”
Maybe post utopianism is an abstract idealisation that makes certain concepts precise.
There are beliefs that directly pay rent, and then there are beliefs that are logical consequences of rent-paying beliefs. The same basic principles that give you applied math will also lead to pure math. We can justify spending effort on pure math on the grounds that it may pay off in the future. However, our belief in pure math is tied to our belief in logic.
If you asked whether this can be applied to something like astrology, I’d ask whether astrology was a logical consequence of beliefs that do pay rent.
Unlike scientific knowledge or other beliefs about the material world, a mathematical fact (e.g. that z follows from X1, X2,..., Xn), once proven, is beyond dispute; there is no chance that such a fact will be contradicted by future observations. One is allowed to believe mathematical facts (once proven) because they are indisputably true; that these facts pay rent is supported by VKS’s argument.
Truths of pure maths don’t pay rent in terms iof expected experience. EY has put forward a criterion of truth, correspondence, and a criterion of believability, expected experience , and pure maths fits neither. He didn’t want that to happen, and the problem remains, here and elsewhere, of how to include abstract maths and still exclude the things you don’t like. This is old ground, that the logical postivists went over in the mid 20th century.
I think I see where you are going with this.
My initial interpretation of EY’s original post is that he was explicating a scientific standard of belief that would make sense in many situations, including in reasoning about the physical world (EY’s initial examples were physical phenomena—trees falling, bowling balls dropping, phlogiston, etc.). I did not really think he was proposing the only standard of belief. This is why I was baffled by your insistence that unless a mathematical fact had made successful predictions about physical, observable phenomena, it should be evicted.
However, later in the original post EY used an example out of literary criticism, and here he appears to be applying the standard to mathematics. So, you may be on to something—perhaps EY did intend the standard to be universally applied.
It seems to me that applying EY’s standard too broadly is tantamount to scientism (which I suspect is more-less the point you were making).
Here is a truth of pure mathematics: every positive integer can be expressed as a sum of four squares.
Expected experiences: there will be proofs of this theorem, proofs that I can follow through myself to check their correctness.
Et voilà!
Truth of astrology: mars in conjunction with Jupiter is dangerous for Leos
Expected experience: there will be astrology articles saying Leo’s are in danger when mars is in conjunction with Jupiter.
Of course astrological claims pay rent. The problem with astrology is not that it’s meaningless but that it’s false, and the problem with astrologers is that they don’t pay the epistemological rent.
Also, a proof is a different thing from a mathematician saying so. The rent that is being paid there is not merely that the theorem will be asserted but that there will be a proof.
Try telling Eliezer
The original post does not mention astrology. If you want to spy out some place where Eliezer has said that astrological claims are meaningless, go right ahead. I am not particularly concerned with whether he has or not.
Here and now, you are talking to me, and as I pointed out, the belief can pay rent, but astrologers are not making it do so. Those who have seriously looked for evidence, have, so I understand, generally found the beliefs false.
I think this is both right and not in contradiction with the post.
The belief that pays the rent here is that there is going to be a high correlation between Mars being in conjunction with Jupiter and astrology believers born around August experiencing heightened feelings of being in danger.
That does not say anything on the “truth” of astrology itself.
Same applies to the article’s example on Wulky Wilkinsen. The belief that alienated resublimation justifies the fictional author’s retropositionality does not pay rent. The belief that failing to mention retropositionality correlates with higher chances of failing a literature test on Wilkinsen does probably pay rent.
From that belief, the expected experience should be Leo people being less fortunate during those days.
That was the point. Its a cheat to expect astrology truths to product experiences of reading written materials about astrology, so it’s a cheat expect to pure maths truths …
Let me complete the ellipsis with what I actually said. A mathematical assertion leads me to expect a proof. Not merely experiences of reading written materials repeating the assertion.
And a proof still isnt an .experience in the relevant sense. Its not like predicting an eclipse,
What’s the difference between behaviours of non-sentient objects and behaviours of sentient people that makes one an experience and the other not?