Can the Sciences Prove that Something is True?

[Polymath257]

That may or may not be the case (there are some pretty strong criticisms of the notion that we cannot know whether or not we are brains in a vat -- but let's not get into those here.), but the problem with certainty in the sciences has more to do with the fundamental nature and limits of inductive logic or reasoning than it does with the brain in a vat notion. Inductive logic (aka reasoning) is fundamental to the sciences since they rely on empirical evidence. However, inductive logic cannot by its very nature provide certainty. Hence, the sciences are ultimately uncertain.

I agree to a large extent. But there are qualifiers.

For example, no observation or measurement has infinite precision. There always *always* error bars around any measurement. But, we can, and do establish that certain results are true *to a certain approximation*. And that is something that science can establish with certainty for each measurement.

So, when we measure the length of a rod, we may not be able to say that the rod is exactly sqrt(2) units long. But we *can* determine with certainty that the rod is between 1.41 and 1.42 units long.

As far as the problem of induction: there is no certainty that the Earth will continue to rotate until tomorrow. But the probability of it failing to happen is so small that we generally ignore it.

I'd also say that you probably over-estimate the 'certainty' that comes from mathematics and logic. Whatever certainty those two subjects have come from them being abstract and having little directly to do with 'reality'. So, we can prove 1+1=2 from very few assumptions, but whether those assumptions apply in nay given physical situation is a completely different matter.

 

[Polymath257]

No, Hilbert's Hotel--in which a fully occupied hotel with infinitely many rooms can still accommodate infinitely many more guests--does not have to be applicable to an actual hotel in order to be true.

But it *does* require some axioms concerning sets and what it means to have an infinite number of rooms.

 

[shunyadragon]

The methods of science are based on what scientists find compelling.

I am uncertain how 'compelling' fits here. The methods of science are like tools in a tool box. Scientist choose the tools that fit the tasks at hand.

They are not derived from philosophy.

Incomplete. Science and math evolved as analytic philosophies.

 

[shunyadragon]

How can the philosophy of science evolve separately from philosophy?

Your statement is incomplete concerning what I posted.

I believe just saying science evolved from philosophy is incomplete and misleading. Specifically science and math evolved as analytic philosophies.

Actually, the basis of math and early fundamentals of physics and chemistry began as practical trial and error analytic philosophy is truly older than classic philosophy, actually in the Neolithic or earlier, and evolved with practical application in the real world to build the pyramids, and great monuments of the world, ship building, military engineering, metallurgy, irrigation, and much more.

 

[shunyadragon]

No, Hilbert's Hotel--in which a fully occupied hotel with infinitely many rooms can still accommodate infinitely many more guests--does not have to be applicable to an actual hotel in order to be true.

No, but if you begin with an empty hotel of infinite rooms, they never can be filled.

 

[Sunstone]

So, when we measure the length of a rod, we may not be able to say that the rod is exactly sqrt(2) units long. But we *can* determine with certainty that the rod is between 1.41 and 1.42 units long.

What we actually establish is that our measurements of the rod up until now indicates that the rod is between 1.41 and 1.42 units long. And by induction, we can reason that any future measurements of the rod will most likely fall within that range. But we cannot say with the same degree of certainty as we can say 2 + 2 = 4 that the rod always will be between 1.41 and 1.42 units long.

As far as the problem of induction: there is no certainty that the Earth will continue to rotate until tomorrow. But the probability of it failing to happen is so small that we generally ignore it.

True, we generally ignore it because we are generally doing things that do not require us to take that degree of uncertainty into account. But in discussing whether the sciences can arrive a certainty, we must either choose to take that degree of uncertainty into account, or to be illogical. One or the other.

I'd also say that you probably over-estimate the 'certainty' that comes from mathematics and logic. Whatever certainty those two subjects have come from them being abstract and having little directly to do with 'reality'. So, we can prove 1+1=2 from very few assumptions, but whether those assumptions apply in nay given physical situation is a completely different matter.

Irrelevant. It makes absolutely no difference in this case whether or not mathematics and deductive logic have anything to do with empirical reality. Their certainty is entirely a matter of logic and does not in any way crucially rely on their correspondence to an empirical reality.

 

[Polymath257]

No, but if you begin with an empty hotel of infinite rooms, they never can be filled.

Sure they can. You have to start with an infinite number of people. If each person is beside a room, then they only have to step forward to fill the room.

Alternatively, take 1 hour to fill the first room, 1/2 hour to fill the second, 1/4 hour to fill the third, etc. After 2 hours, the rooms are full!

 

[Sunstone]

I am uncertain how 'compelling' fits here. The methods of science are like tools in a tool box. Scientist choose the tools that fit the tasks at hand.

I think it would be more accurate to say scientists choose or create the methods that they find provide compelling reasons and evidence for their notions. Put more concretely, if you were a scientist, you would discard methods that no one found compelling and retain and use methods that could convince you and your colleagues.

Incomplete. Science and math evolved as analytic philosophies.

Please provide some cases in which science and math evolved as analytic philosophies. But please do not simply name them, as that demonstrates absolutely nothing. Instead, please show in sufficient detail exactly what their evolution was.

 

[Polymath257]

What we actually establish is that our measurements of the rod up until now indicates that the rod is between 1.41 and 1.42 units long. And by induction, we can reason that any future measurements of the rod will most likely fall within that range. But we cannot say with the same degree of certainty as we can say 2 + 2 = 4 that the rod always will be between 1.41 and 1.42 units long.

True, we generally ignore it because we are generally doing things that do not require us to take that degree of uncertainty into account. But in discussing whether the sciences can arrive a certainty, we must either choose to take that degree of uncertainty into account, or to be illogical. One or the other.

And I agree that in that sense, there is no certainty in the real world.

Irrelevant. It makes absolutely no difference in this case whether or not mathematics and deductive logic have anything to do with empirical reality. Their certainty is entirely a matter of logic and does not in any way crucially rely on their correspondence to an empirical reality.

On the contrary, you have to consider the possibility that our brains will assent to an argument even if that argument is wrong. So the simple fact that we can prove 1+1=2 from certain axioms assumes that we can detect a false argument involving those assumptions. In this way, even our certainty of math and logic is limited.

\E: Essentially, we have to assume that we are sane in order to have confidence in our logic and math.

 

[Sunstone]

On the contrary, you have to consider the possibility that our brains will assent to an argument even if that argument is wrong. So the simple fact that we can prove 1+1=2 from certain axioms assumes that we can detect a false argument involving those assumptions. In this way, even our certainty of math and logic is limited.

That's fascinating! I'm going to need to think about that one. Thanks!

 

[shunyadragon]

Sure they can. You have to start with an infinite number of people. If each person is beside a room, then they only have to step forward to fill the room.

Alternatively, take 1 hour to fill the first room, 1/2 hour to fill the second, 1/4 hour to fill the third, etc. After 2 hours, the rooms are full!

Disagree, if you consider the infinite number of empty hotel rooms as potential infinity there will always be another hotel room no matter how many people are waiting to fill the rooms.

Maybe if you are describing the hotel in terms of an actual infinity, but . . .

. . . potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps.
Actual infinity - Wikipedia

 

[Polymath257]

That's fascinating! I'm going to need to think about that one. Thanks!

I can go farther. There are many examples in the mathematics literature where people believed that they had a proof of a result, but it turns out that they made a mistake and the proof fell through. For some, the end result turned out to be false.

So even mathematical proof, as it is done by humans, is uncertain. I can almost guarantee that there are wrong results in many top-notch journals. of course, without going through each proof and finding the mistake, I can't say which ones are wrong. But.....

 

[Nous]

But it *does* require some axioms concerning sets and what it means to have an infinite number of rooms.

Yes, agree. Obviously that doesn't mean that mathematics isn't true unless it is applicable to something empirical.

 

[Polymath257]

Disagree, if you consider the infinite number of empty hotel rooms as potential infinity there will always be another hotel room no matter how many people are waiting to fill the rooms.

Maybe if you are describing the hotel in terms of an actual infinity, but . . .

. . . potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps.
Actual infinity - Wikipedia

I was certainly thinking in terms of actual infinities. Potential infinities are just limits in disguise.

 

[Polymath257]

Yes, agree. Obviously that doesn't mean that mathematics isn't true unless it is applicable to something empirical.

I would hesitate the use the word 'true'. I would use 'proven' since that is more applicable to an axiomatic system.

 

[Sunstone]

I can go farther. There are many examples in the mathematics literature where people believed that they had a proof of a result, but it turns out that they made a mistake and the proof fell through. For some, the end result turned out to be false.

So even mathematical proof, as it is done by humans, is uncertain. I can almost guarantee that there are wrong results in many top-notch journals. of course, without going through each proof and finding the mistake, I can't say which ones are wrong. But.....

I understand what you're saying, but isn't that a separate problem from the question of whether we can (at least in theory) know that something is certain? I mean I think you're getting at a practical problem, rather than a theoretical.

 

[Nous]

No, but if you begin with an empty hotel of infinite rooms, they never can be filled.

Polymath's reply is better than I could have done.

 

[Polymath257]

I understand what you're saying, but isn't that a separate problem from the question of whether we can (at least in theory) know that something is certain? I mean I think you're getting at a practical problem, rather than a theoretical.

Can we know for certain that our minds can detect false proofs?

 

[Nous]

I would hesitate the use the word 'true'. I would use 'proven' since that is more applicable to an axiomatic system.

Agree.

 

[Sunstone]

Can we know for certain that our minds can detect false proofs?

I'm still thinking about that one. On the surface, the answer is "no", but I want to think it through a bit more.