Comment author:VKS
17 March 2012 03:35:25PM
27 points
[-]

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.

Comment author:jirkazr
23 August 2012 02:51:29PM
*
5 points
[-]

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.

Comment author:g_pepper
22 February 2015 10:12:05PM
0 points
[-]

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.

Comment author:lalaithion
22 February 2015 11:15:06PM
1 point
[-]

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.

Comment author:g_pepper
22 February 2015 11:58:10PM
*
0 points
[-]

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.

Comment author:g_pepper
24 February 2015 01:07:30AM
*
1 point
[-]

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).

Comment author:Epictetus
23 February 2015 06:07:49AM
0 points
[-]

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.

Comment author:Epictetus
23 February 2015 04:22:00PM
1 point
[-]

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.

## Comments (245)

OldWhat 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.

*5 points [-]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 :-)

Is pure math a set of beliefs that should be evicted?

No, for reasons expressed above by VKS.

*0 points [-]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.

*0 points [-]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

allowedto believe mathematical facts (once proven) because they are indisputably true; that these factspay rentis supported by VKS's argument.*1 point [-]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.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.

*1 point [-]I think I see where you are going with this.

My initial interpretation of EY's original post is that he was explicating a

scientificstandard 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 proposingthe onlystandard 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

didintend 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).

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.

*0 points [-]I am not making the

statement"exclude pure math", I am posing thequestion"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 efforton pure math on the grounds that it may pay off in the future. However, ourbeliefin 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.