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It is said that in the real world nothing can be proven to be absolutely true because there is always a way for a statement to be disproved by a counterargument. For example, a person "A" could say that the apple felt down from a tree because that person "A" saw it. That person would also label his statement as the true statement. Other person "B" could argue that the person "A" has no way to prove its statement to be true and furthermore the person "B" can disprove such a statement by saying for example: "You cannot disprove the fact that your memory of that moment of an apple falling down was just put in to your brain by some evil Got who is manipulating with your memories." (We could also use some other sceptical hypothesis for disproving such a statement)

Therefore in this sense it seems to me that people are only capable of knowing the truth which they assume to be true. Also they can derive other truths from such assumed truths (we could lable such believed truths as axioms or assumptions) by some inference rules which they assume to preserve the truth value of those axioms. Therefore everything that people lable as a true statement is just a belief because they cannot know for sure (they aren't able to prove it and it is always possible to disprove it). Therefore in this sense a scientific truth is equal to a mathematical truth if we believe that both of them are true.

From this perspective it seems to me impossible for a person to know any other kind of truth than the one that he believes in. Is it therefore reasonable for a person to argue that a scientific truth and a mathematical truth are equal because the only difference is in the things person believes are true?

What are some viewpoints on this matter in philosophy?

(Please try to write in a language understandable to a person who doesn't have a philosophical nor logical background)

  • About "the things person believes are true" you can compare the belief that we (humans) can move ourselves using only the "power of the mind" (which is clearly false) with the mechanics used to land on the Moon (a scientific theory that works). – Mauro ALLEGRANZA Aug 25 at 14:27
  • This might work for a radical skeptic, but they are not considered reasonable. The rest of us make finer distinctions. In particular, something that depends only on our conventions (like math) may still be misjudged due to oversight. But no amount of care can prevent a mistake in science, because the reality may present us with something we never encountered before. It is reasonable to distinguish what we can control from what we can not, after all, our survival depends on it. – Conifold Aug 26 at 6:07
  • To a question of similar nature some thoughts of mine here. – Rusi-packing-up Aug 26 at 6:52
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There is no difficulty in assuming that we don't actually know anything about the material world. We don't even need to claim to know that we know nothing about it since we may believe it is the case, or indeed believe it is not the case.

It won't make any difference if we can only have beliefs about the material world and it won't make any difference if we can know the material world but are unable to tell the difference between belief and knowledge.

Our possible claims to knowledge can be seen as just beliefs we believe are knowledge. Beliefs we think of as knowledge but are not can be regarded as beliefs free from doubt.

And I don't see why they would have to be any substantial difference in behaviour between people who believe they know and people who believe they believe. I certainly seem myself to keep behaving as if I knew my way around even though I believe that I actually don't.

We can think of human beings as acting according to whatever they happen to believe and we can think of scientific theories as particular beliefs. We don't even need to pretend that scientific theories are knowledge since it is good enough to believe they are true. And that we act accordingly.

There is also no difficulty is ranking our beliefs in terms of the kind of justification we think we have to have them. And we can rank science as our most certain belief about the material world. I certainly do.

And we can also rate our own perception of the material world as more reliable most of the time than anything other people might want to claim they know. And I certainly do.

Is mathematics any different?

Obviously, we think of mathematics as an abstraction, and apparently even most mathematicians do. However, there is also no difficulty if we accept that no mathematician knows any mathematical truth.

As in the case of our beliefs about the material world, it is enough to think of mathematicians as acting according to their beliefs about mathematical truths. Here again, there is no substantial difference between believing you know and believing you believe as long as you act according to your belief, which seems to be what the notion of belief suggests we do anyway.

Knowledge of the material world is similar to the notion of the infinite. It makes no practical difference as long as you don't claim to know that you know. Claiming to know that you know does expose you to the risk of being contradicted by the facts of the matter, or at least by something you may come to believe about them at some point in the future.

Why take that risk? Well, we believe it can be very useful. We believe we can leverage our claim to have something we don't have to gain some substantial material advantages. And we all do it, anyway (I believe).

British mathematician Andrew Wiles may believe he proved Fermat's Last Theorem and maybe he did, and that he did or did not would make a substantial difference, but that he merely believes he did, and that we all believe he did, instead of all believing that we know he did, doesn't make any substantial difference.

Just like when I say "I love you". What matters is not that we should know it to be true, but that we should all believe it to be true, and then not even forever, but just on the moment.

And act on it.

Which explains gracefully why we seem to be here at all to tell the tale.

There is no good reason to assume any substantial difference between our various senses of perception of the material world, on the one hand, and our mental capabilities, such as memory, logical sense, and even feelings and abstract thought, on the other. The latter capabilities are essentially, in all but name, perception of the material world, with the caveat that they are perception of that part of the material we call our body, and concerning our memory and our logic, that part of the material world we call our brain.

Thus, mathematical abstractions behave in every way as do anything in the material world. We perceive them where they are, in this case in our own brain and they appear to us just as characteristically themselves as do the tree in my garden. Differences are just as significant as differences between vision and hearing, for example.

Thus, mathematical abstractions may just be as unknowable as anything else. You know the one you have in mind, but then you can't hold in mind all mathematical abstractions at the same time. And there goes our claim to knowledge down the drain.

Further, since physics is now suggesting a universe which many people like to think of as essentially mathematical--that is, somehow mathematical in nature, whatever that may mean exactly--there is a near perfect analogy between knowledge of the material world and knowledge of the Platonic world of mathematical abstractions.

We are able to believe we know them.

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