At any rate, if the former is true, 2+2=4 is outside the province of empirical science, and applying empirical reasoning to evaluate its 'truth' is wrong.

When I imagine putting two apples next to two apples, I can predict what will actually happen when I put two earplugs next to two earplugs, and indeed, my mind can store the result in a generalized fashion which makes predictions in many specific instances. If you do not call this useful abstract belief "2 + 2 = 4", I should like to know what you call it. If the belief is outside the province of empirical science, I would like to know why it makes such good predictions.

To apply the same reasoning the other way, if you aren't a Christian, what would be a situation which would convince you of the truth of Christianity?

You'd have to fix all the problems in belief, one by one, by reversing the evidence that originally convinced me of the beliefs' negations. If the Sun stopped in the sky for a day, and then Earth's rotation restarted without apparent damage, that would convince me there was one heck of a powerful entity in the neighborhood. It wouldn't show the entity was God, which would be much more complicated, but it's an example of how one small piece of my model could be flipped from the negation of Christianity (in that facet) to the non-negation.

Getting all the pieces of the factual model (including the parts I was previously convinced were logically self-contradictory) to align with Christianity's factual model, would still leave all the ethical problems. So the actual end result would be to convince me that the universe was in the hands of a monstrously insane and vicious God. But then there does not need to be any observable situation which convinces me that it is morally acceptable to murder the first-born children of Egyptians - morality does not come from environmental entanglement.

Comment author:tut
15 July 2009 01:10:15PM
14 points
[-]

If you do not call this useful abstract belief "2 + 2 = 4", I should like to know what you call it.

I call it "2+2=4 is a useful model for what happens to the number of earplugs in a place when I put two earplugs beside two other earplugs". Which is a special case of the theory "arithmetic is a useful model for numbers of earplugs under some operations (including but not limited to adding and removing)".

If the belief is outside the province of empirical science, I would like to know why it makes such good predictions.

The mathematical claim "2+2=4" makes no predictions about the physical world. For that you need a physical theory. 2+2=4 would be true in number theory even if your apples or earplugs worked in some completely different manner.

Comment author:bigjeff5
03 March 2011 06:53:16PM
19 points
[-]

I hate to break it to you, but if setting two things beside two other things didn't yield four things, then number theory would never have contrived to say so.

Numbers were invented to count things, that is their purpose. The first numbers were simple scratches used as tally marks circa 35,000 BC. The way the counts add up was derived from the way physical objects add up when grouped together. The only way to change the way numbers work is to change the way physical objects work when grouped together. Physical reality is the basis for numbers, so to change number theory you must first show that it is inconsistent with reality.

Thus numbers have a definite relation to the physical world. Number theory grew out of this, and if putting two objects next to two other objects only yielded three objects when numbers were invented over forty thousand years ago, then number theory must reflect that fact or it would never have been used. Consequently, suggesting 2+2=4 would be completely absurd, and number theorists would laugh in your face at the suggestion. There would, in fact, be a logical proof that 2+2=3 (much like there is a logical proof that 2+2=4 in number theory now).

All of mathematics are, in reality, nothing more than extremely advanced counting. If it is not related to the physical world, then there is no reason for it to exist. It follows rules first derived from the physical world, even if the current principles of mathematics have been extrapolated far beyond the bounds of the strictly physical. I think people lose sight of this far too easily (or worse, never recognize it in the first place).

Mathematics are so firmly grounded in the physical reality that when observations don't line up with what our math tells us, we must change our understanding of reality, not of math. This is because math is inextricably tied to reality, not because it is separate from it.

In your last paragraph you turn everything around and inexplicably claim that math is more primary than observation of reality, though you did a good job -- and one I agree with -- of pointing out the opposite in the previous part of the comment.

Comment author:bigjeff5
04 March 2011 04:12:24AM
9 points
[-]

When it was noticed in the 1800's that the perihelion of Mercury did not match what Newton's inverse-square law of gravity predicted, did we change the way math works? Or did we change our understanding of gravity?

Math is the most fundamental understanding of reality that we have. It is the most thoroughly supported and proven aspect of science that I know of. That doesn't mean that our understanding of math can't be fundamentally flawed, but it does mean that math is the last place we expect to find a problem when our observations don't match our expectations.

In other words, when assigning probabilities to whether math is wrong or Newton's Theory of Gravity is wrong, the probability we assign to math itself being wrong is something like 0.000001% (sorry, I don't know nearly enough math to make it less than that) and Newton's Gravity being wrong something like 99.999999%.

You're saying that in the mid nineteenth century (half a century before relativity), the anomalous precession of Mercury made it seem 99.999999% likely that Newtonian mechanics was wrong?

After all, there are other possibilities.

cf. "When it was noticed in the 1800's that the perihelion of Neptune did not match what Newton's inverse-square law of gravity predicted, did we change the way math works? Or did we change our understanding of gravity?"
In this case we actually postulated the existence of Pluto.

Similar solutions were suggested for the Mercury case, e.g. an extremely dense, small object orbiting close to Mercury.

If I were a nineteenth century physicist faced with the deviations in the perihelion of Mercury, I'd give maybe a 0.1% probability to Newton being incorrect, a 0.001% probability to maths being incorrect, and the remaining ~99.9% would be shared between incorrect data /incomplete data/ other things I haven't thought of.

However, I agree that we can probably be more confident of results in maths than results in experimental science. (I was going to distinguish between mathematical/empirical results, but given that the OP was to do with the empirical confirmation of maths, I thought "mathematical/experimental" would be a safer distinction)

Comment author:elharo
23 March 2013 12:07:15PM
*
2 points
[-]

For well-established math, sure. We certainly will look for experimental mistakes, unnoticed observables (e.g. the hypothesized planet Vulcan to explain Mercury's deviation from Newtonian gravity), and better theories in about that order. However for less well established mathematics at the frontiers we do consider the possibility that we've made a mistake somewhere.

Off the top of my head the biggest example I can think of was von Neumann's proof that hidden variables were inconsistent with quantum mechanics, which was widely believed and cited at least into the 1980s, despite the fact that David Bohm published a consistent hidden variables theory of quantum mechanics in 1952. I'm curious if anyone can recall a case in which an experimental result led us to realize that a previously accepted mathematical "fact" was incorrect.

Comment author:matteyas
09 October 2014 12:42:25AM
1 point
[-]

I hate to break it to you, but if setting two things beside two other things didn't yield four things, then number theory would never have contrived to say so.

At what point are there two plus two things, and at what point are there four things? Would you not agree that a) the distinction itself between things happens in the brain and b) the idea of the four things being two separate groups with two elements each is solely in the mind? If not, I'd very much like to see some empirical evidence for the addition operation being carried out.

Mathematics are so firmly grounded in the physical reality that when observations don't line up with what our math tells us, we must change our understanding of reality, not of math.

English is so firmly grounded in the physical reality that when observations don't line up with what our english tells us, we must change our understanding of reality, not of english.

I hope the absurdity is obvious, and that there are no problems to make models of the world with english alone. So, do you find it more likely that math is connected to the world because we link it up explicitly or because it is an intrinsic property of the world itself?

Comment author:Peterdjones
09 October 2014 03:32:14AM
0 points
[-]

Mathematics are so firmly grounded in the physical reality that when observations don't line up with what our math tells us, we must change our understanding of reality, not of math. This is because math is inextricably tied to reality, not because it is separate from it.

Comment author:Lethalmud
11 June 2013 01:42:18PM
*
0 points
[-]

Elezier, do you believe that someday humans could create an AI and put that AI in a simulated enviroment that accurately simulated all the observations humanity made until now?

If you do, what further observations would that AI have to make to arrive at the belief that they were created by an intelligent entity?

Comment author:khafra
12 June 2013 05:42:16PM
3 points
[-]

If we assume that humanity has gained access to effectively infinite computing power, and has put AIXItl or something similar into a copy of the universe, simulated at whatever level unifies quantum mechanics and gravitation into a coherent, leakproof framework, AIXItl would have an extremely small belief that it was inside a simulation. Only if the simplest unification of quantum mechanics and gravity turns out to be "we're in a simulation," would a hyperintelligent AI in a perfect simulation of our universe come to the belief that it's in a simulation.

So, the epistemically perfect AI would come to an incorrect decision. This does not imply a flaw in its method for forming beliefs; it merely implies the tautology that there is no way to find out what there is no way to find out.

When I imagine putting two apples next to two apples, I can predict what will actually happen when I put two earplugs next to two earplugs, and indeed, my mind can store the result in a generalized fashion which makes predictions in many specific instances. If you do not call this useful abstract belief "2 + 2 = 4", I should like to know what you call it. If the belief is outside the province of empirical science, I would like to know why it makes such good predictions.

No, the real world does not work via Peano arithmetic. Your experiments with apples and earplugs are simply applications of conservation of mass and immutability of inanimate objects, and other such principles. Before you learned such things, you were thrilled with the game of peek-a-boo -- of how someone could cease to exist, and then appear out of nowhere.

Consider this experiment: Take 2 apples, cut them in half. Take 2 more apples, cut them in half. Put all together. How many apples do you have? The answer is not "4 apples", the answer is "8 half-apples". Furthermore, each individual apple remains the same apple as before (minus the effects of time), so that any differences in size, shape, coloration, bruising, etc would remain the same. Apples aren't numbers, and can't be substituted for each other.

The world abounds with examples where Peano arithmetic does not apply. Consider adding two speeds together -- they do not add via Peano arithmetic, such that there exists a speed X, such that 2X + 2X = 3X. If we're using naive multiplication, that speed is where X=c*sqrt(7/36). None of this changes my beliefs about Peano arithmetic -- it is necessarily true given its axioms, and its correspondence to the physical world is entirely coincidental. Certainly, if Peano arithmetic didn't correspond widely to real world problems, I would never have learned about it in high school and it might not even have been invented -- but it remains true all the same.

This all just means that my idea of truth is different than yours -- I think things can be true or false regardless of their predictive value. Specifically, I value statements of the form "If A, then (A worded slightly differently)" and think that almost all knowledge has that form. For example, "If the universe is consistent and objective, the scientific method will tend toward accurately describing the universe". Once you introduce inductive reasoning, even for something as trivial as stating "the universe is consistent and objective", then you introduce uncertainty -- you switch from binary true/false to likely/unlikely and accurate/inaccurate and predictive/non-predictive.

If you equate "scientific type truth" with other people's "actual truth" you will get into many pointless arguments. For example, you seem greatly offended that religious people sometimes disagree with science. For example, they will point out that science has always been wrong, and is probably still wrong. But they'll probably agree that modern science more accurately and more precisely predicts about our world. In fact, you'll probably get them to agree that science is perhaps the best method to make accurate and precise predictions. A statement doesn't need to be true to be useful for making accurate predictions, for example Newtonian gravity. So why equate "makes accurate predictions" with "truth"?

## Comments (390)

OldAt any rate, if the former is true, 2+2=4 is outside the province of empirical science, and applying empirical reasoning to evaluate its 'truth' is wrong.When I imagine putting two apples next to two apples, I can predict what will actually happen when I put two earplugs next to two earplugs, and indeed, my mind can store the result in a generalized fashion which makes predictions in many specific instances. If you do not call this useful abstract belief "2 + 2 = 4", I should like to know what you call it. If the belief is outside the province of empirical science, I would like to know why it makes such good predictions.

To apply the same reasoning the other way, if you aren't a Christian, what would be a situation which would convince you of the truth of Christianity?You'd have to fix all the problems in belief, one by one, by reversing the evidence that originally convinced me of the beliefs' negations. If the Sun stopped in the sky for a day, and then Earth's rotation restarted without apparent damage, that would convince me there was one heck of a powerful entity in the neighborhood. It wouldn't show the entity was God, which would be much more complicated, but it's an example of how one small piece of my model could be flipped from the negation of Christianity (in that facet) to the non-negation.

Getting all the pieces of the factual model (including the parts I was previously convinced were logically self-contradictory) to align with Christianity's factual model, would still leave all the ethical problems. So the actual end result would be to convince me that the universe was in the hands of a monstrously insane and vicious God. But then there does not need to be any observable situation which convinces me that it is morally acceptable to murder the first-born children of Egyptians - morality does not come from environmental entanglement.

I call it "2+2=4 is a useful model for what happens to the number of earplugs in a place when I put two earplugs beside two other earplugs". Which is a special case of the theory "arithmetic is a useful model for numbers of earplugs under some operations (including but not limited to adding and removing)".

The mathematical claim "2+2=4" makes no predictions about the physical world. For that you need a physical theory. 2+2=4 would be true in number theory even if your apples or earplugs worked in some completely different manner.

I hate to break it to you, but if setting two things beside two other things didn't yield four things, then number theory would never have contrived to say so.

Numbers were invented to count things, that is their purpose. The first numbers were simple scratches used as tally marks circa 35,000 BC. The way the counts add up was derived from the way physical objects add up when grouped together. The only way to change the way numbers work is to change the way physical objects work when grouped together. Physical reality is the basis for numbers, so to change number theory you must first show that it is inconsistent with reality.

Thus numbers have a definite relation to the physical world. Number theory grew out of this, and if putting two objects next to two other objects only yielded three objects when numbers were invented over forty thousand years ago, then number theory must reflect that fact or it would never have been used. Consequently, suggesting 2+2=4 would be completely absurd, and number theorists would laugh in your face at the suggestion. There would, in fact, be a logical proof that 2+2=3 (much like there is a logical proof that 2+2=4 in number theory now).

All of mathematics are, in reality, nothing more than extremely advanced counting. If it is not related to the physical world, then there is no reason for it to exist. It follows rules first derived from the physical world, even if the current principles of mathematics have been extrapolated far beyond the bounds of the strictly physical. I think people lose sight of this far too easily (or worse, never recognize it in the first place).

Mathematics are so firmly grounded in the physical reality that when observations don't line up with what our math tells us, we must change our understanding of reality, not of math. This is because math is inextricably tied to reality, not because it is separate from it.

*7 points [-]Verbal expressions almost certainly predate physical notations. Unfortunately the echos don't last quite that long.

In your last paragraph you turn everything around and inexplicably claim that math is more primary than observation of reality, though you did a good job -- and one I agree with -- of pointing out the opposite in the previous part of the comment.

When it was noticed in the 1800's that the perihelion of Mercury did not match what Newton's inverse-square law of gravity predicted, did we change the way math works? Or did we change our understanding of gravity?

Math is the most fundamental understanding of reality that we have. It is the most thoroughly supported and proven aspect of science that I know of. That doesn't mean that our understanding of math can't be fundamentally flawed, but it does mean that math is the last place we expect to find a problem when our observations don't match our expectations.

In other words, when assigning probabilities to whether math is wrong or Newton's Theory of Gravity is wrong, the probability we assign to math itself being wrong is something like 0.000001% (sorry, I don't know nearly enough math to make it less than that) and Newton's Gravity being wrong something like 99.999999%.

See what I'm saying?

Yup. I think we agree. My disagreeing post was a mere misunderstanding of what you were saying.

After a few recent posts of mine it looks like I need to work on my phrasing in order to make my points clear.

No harm no foul.

Woah, I think that's a little overconfident...

You're saying that

in the mid nineteenth century(half a century before relativity), the anomalous precession of Mercury made it seem 99.999999% likely that Newtonian mechanics was wrong?After all, there are other possibilities.

cf. "When it was noticed in the 1800's that the perihelion of Neptune did not match what Newton's inverse-square law of gravity predicted, did we change the way math works? Or did we change our understanding of gravity?" In this case we actually postulated the existence of Pluto.

Similar solutions were suggested for the Mercury case, e.g. an extremely dense, small object orbiting close to Mercury.

And that's leaving aside the fact that 99.999999% is an absurdly high level of confidence for pretty much any statement at all (see http://lesswrong.com/lw/mo/infinite_certainty/ ).

If I were a nineteenth century physicist faced with the deviations in the perihelion of Mercury, I'd give maybe a 0.1% probability to Newton being incorrect, a 0.001% probability to maths being incorrect, and the remaining ~99.9% would be shared between incorrect data /incomplete data/ other things I haven't thought of.

However, I agree that we can probably be more confident of results in maths than results in experimental science. (I was going to distinguish between mathematical/empirical results, but given that the OP was to do with the empirical confirmation of maths, I thought "mathematical/experimental" would be a safer distinction)

*2 points [-]For well-established math, sure. We certainly will look for experimental mistakes, unnoticed observables (e.g. the hypothesized planet Vulcan to explain Mercury's deviation from Newtonian gravity), and better theories in about that order. However for less well established mathematics at the frontiers we do consider the possibility that we've made a mistake somewhere.

Off the top of my head the biggest example I can think of was von Neumann's proof that hidden variables were inconsistent with quantum mechanics, which was widely believed and cited at least into the 1980s, despite the fact that David Bohm published a consistent hidden variables theory of quantum mechanics in 1952. I'm curious if anyone can recall a case in which an experimental result led us to realize that a previously accepted mathematical "fact" was incorrect.

Here's a whole gallery of math which we were later proven to be mistaken about.

At what point are there two plus two things, and at what point are there four things? Would you not agree that a) the distinction itself between things happens in the brain and b) the idea of the four things being two separate groups with two elements each is solely in the mind? If not, I'd very much like to see some empirical evidence for the addition operation being carried out.

English is so firmly grounded in the physical reality that when observations don't line up with what our english tells us, we must change our understanding of reality, not of english.

I hope the absurdity is obvious, and that there are no problems to make models of the world with english alone. So, do you find it more likely that math is connected to the world because we link it up explicitly or because it is an intrinsic property of the world itself?

On the other hand...

http://en.m.wikipedia.org/wiki/Is_logic_empirical%3F

*0 points [-]Elezier, do you believe that someday humans could create an AI and put that AI in a simulated enviroment that accurately simulated all the observations humanity made until now?

If you do, what further observations would that AI have to make to arrive at the belief that they were created by an intelligent entity?

If we assume that humanity has gained access to effectively infinite computing power, and has put AIXI

tlor something similar into a copy of the universe, simulated at whatever level unifies quantum mechanics and gravitation into a coherent, leakproof framework, AIXItlwould have an extremely small belief that it was inside a simulation. Only if the simplest unification of quantum mechanics and gravity turns out to be "we're in a simulation," would a hyperintelligent AI in a perfect simulation of our universe come to the belief that it's in a simulation.So, the epistemically perfect AI would come to an incorrect decision. This does not imply a flaw in its method for forming beliefs; it merely implies the tautology that there is no way to find out what there is no way to find out.

No, the real world does not work via Peano arithmetic. Your experiments with apples and earplugs are simply applications of conservation of mass and immutability of inanimate objects, and other such principles. Before you learned such things, you were thrilled with the game of peek-a-boo -- of how someone could cease to exist, and then appear out of nowhere.

Consider this experiment: Take 2 apples, cut them in half. Take 2 more apples, cut them in half. Put all together. How many apples do you have? The answer is not "4 apples", the answer is "8 half-apples". Furthermore, each individual apple remains the same apple as before (minus the effects of time), so that any differences in size, shape, coloration, bruising, etc would remain the same. Apples aren't numbers, and can't be substituted for each other.

The world abounds with examples where Peano arithmetic does not apply. Consider adding two speeds together -- they do not add via Peano arithmetic, such that there exists a speed X, such that 2X + 2X = 3X. If we're using naive multiplication, that speed is where X=c*sqrt(7/36). None of this changes my beliefs about Peano arithmetic -- it is necessarily true given its axioms, and its correspondence to the physical world is entirely coincidental. Certainly, if Peano arithmetic didn't correspond widely to real world problems, I would never have learned about it in high school and it might not even have been invented -- but it remains true all the same.

This all just means that my idea of truth is different than yours -- I think things can be true or false regardless of their predictive value. Specifically, I value statements of the form "If A, then (A worded slightly differently)" and think that almost all knowledge has that form. For example, "If the universe is consistent and objective, the scientific method will tend toward accurately describing the universe". Once you introduce inductive reasoning, even for something as trivial as stating "the universe is consistent and objective", then you introduce uncertainty -- you switch from binary true/false to likely/unlikely and accurate/inaccurate and predictive/non-predictive.

If you equate "scientific type truth" with other people's "actual truth" you will get into many pointless arguments. For example, you seem greatly offended that religious people sometimes disagree with science. For example, they will point out that science has always been wrong, and is probably still wrong. But they'll probably agree that modern science more accurately and more precisely predicts about our world. In fact, you'll probably get them to agree that science is perhaps the best method to make accurate and precise predictions. A statement doesn't need to be true to be useful for making accurate predictions, for example Newtonian gravity. So why equate "makes accurate predictions" with "truth"?