Evidently something is going pretty badly wrong here. Here's a truth: my laptop computer is right now on my lap as I'm typing this. It doesn't need "precise accuracy without error for an infinite amount of time" to establish that. It's a rough-and-ready proposition about the here-and-now: precise accuracy and infinite amounts of time just don't come into it. Likewise for many common-or-garden truths.

Something else is going badly wrong. For here's another truth: it rained here today. Nothing there about the human condition and human behaviour. Just a local meteorological fact.

Getting serious about philosophy is nothing to do with "loving a good argument", or trying to make up definitions off the top of your head, any more than getting serious about physics is. It's hard work, and you need to do your homework first in either case. Try this article on truth as a starting point, or Simon Blackburn's Truth.

It is a requirement for something to be genuinely known to be true that it is true. So knowledge implies truth (in the sense that if X knows that so-and-so, then it is the case that so-and-so).

But that doesn't make knowledge the same as truth. The implication the other way around doesn't hold. There are truths that you don't know, that I don't know, and indeed that no one right now knows (maybe because nobody has bothered to find them out, maybe because the time has past when anyone could check, or because the truths are about far-off events like meteorite strikes on the far side of the moon, or for other kinds of reason).

Leaving ominiscient deities out of it, not every truth is known. But we might wonder whether every truth is knowable, in principle, e.g. by a suitably placed and sufficiently smart observer. The trouble with that idea is in spelling out the "in principle".

You write that facts and ideas are very different things. (You also contrast beliefs and facts to make the same point, so perhaps you believe that beliefs are ideas.) From this you infer a difficulty about the possibility of ideas and facts corresponding to one another. 'Facts and ideas are very different things', you write, so 'how can they "correspond" to each other?' Consider, though. Written notes on a stave are very different from the sounds that we hear, but why should that stop them "corresponding" to sounds? Aunts and nephews are very different kinds of beings, but that need not stop them corresponding. You put "correspond" in scare quotes, and here you seem to me to be on the right track. We need to know what correspondence is. What is say a 1:1 correspondence?

What does it meant to say that a belief is true if it corresponds to a fact? A low-calorie reading of that slogan is that my belief e.g. that snow is white is true if and only if it is a fact that snow is white, i.e. if and only if snow is white. Similarly for other beliefs. And there doesn't seem to be anything very mysterious about this claim. (Nor is there much reason to suppose that "most people" are committed to any more.)

Now, it's not that there aren't interesting problems hereabouts. But as the great Cambridge philosopher Frank Ramsey noted, the serious problems are about the nature of "intentionality" or "aboutness". For how can a state of me somehow be about something else, in this case, the colour of snow? (Note, however, the problem isn't in general one about relating separate realms -- I and my states are part of the same natural world as colours and snow! And there are naturalistic stories on the market which tell us how the link-up is made.)

But once we've explained how it is that my belief is about snow's being white, the suggestion is that there isn't a further big problem about explaining what it is for my belief to be true -- to repeat, it will just be true if snow is white.

(OK, I'm fibbing. Getting the minimalist story to fly, steering round the Liar Paradox and other pitfalls, is actually no easy task. The devil is in the details. But I don't think the logical intricacies of the details matter to the general line of response to the broad metaphysical question as posed.)

Perhaps there are two different questions here. There's a very general question about truth and proof; and there's a much more specific question about the sort of case exemplified by Euler, where a mathematician claims to know (or at least have good grounds for) a proposition even in the absence of a demonstrative proof.

Let's take the specific question first, using a different and perhaps more familiar example. We don't know how to prove Goldbach's conjecture that every even number greater than two is the sum of two primes. Yet most mathematicians are pretty confident in its truth. Why?

Well, it has been computer-verified for numbers up to the order of 10¹⁶. But so what? After all, there are other well-known cases where a property holds of numbers up to some much greater bound but then fails. [For example, the logarithmic integral function li(n) over-estimates the number of primes below n but eventually under-estimates, then over-estimates again, flipping back and forth, with the first tipping point now thought to be in the order of 10³¹⁶. See here.] We know, then, that extrapolation from (merely!) the first 10¹⁶ cases is dangerous -- for that "sample" is biased towards relatively tiny numbers. Yet, as I said, mathematicians do all the same tend to be confident in Goldbach's conjecture. Which suggests that they have other non-demonstrative grounds for their belief, grounds better than mere extrapolation from the initial cases. For a discussion of what these might be, and some evaluative comments, see e.g. Alan Baker's paper "Is there a problem of induction for mathematics?" in M. Leng, A. Paseau & M. Potter (eds.) Mathematical Knowledge. That's a pretty accessible read, and one of the few recent papers I know about the interesting question of the role of non-demonstrative reasoning in mathematical thought.

Now, suppose NM (for "naive mathematician"!) does believe Goldbach's conjecture just on the basis of extrapolation from small cases. Then, if a proof were discovered, could NM claim to have known the conjecture to be true already? Surely not so. NM's mathematical reason for his belief was (demonstrably!) a bad reason; and a true belief held for a bad reason isn't a case of genuine knowledge (it's more like a lucky guess). So actually, I think it is wrong in general to say "if you tell me you know that a statement is true, and then in fact someone later proves it, I can't show mathematically that you didn't know it all along." NM didn't know Goldbach's conjecture all along, and that's because his grounds don't pass muster as good mathematical reasons.

Now a nod towards the much more general question here. Certainly, it is very plausible to say that when we prove a mathematical proposition we aren't (so to speak) creating a new truth, but discovering something that was true all along -- "a mathematical statement is true whether we know it or not". Even when mathematicians invent a new branch of mathematics -- as Eilenberg and Mac Lane did in founding category theory -- it remains tempting to say that they are discovering pre-existent patterns and structures. But is this right? For what is the nature of this supposed realm of pre-existing mathematical structures, and how do we get epistemic access to it? Well, they are two of the Big Questions in the philosophy of mathematics. And here, I can probably do no better than just point to Stewart Shapiro's fine introductory book, Thinking About Mathematics.

Yes, at least theoretically. An example of how this might be is given in the first of Descartes' Meditations on First Philosophy. Descartes asks us to consider a world that is governed by a kind of evil god who delights in nothing more than making us believe what is false. In such a world, we would be able to find no evidence at all to debunk the falsehoods to which the god inclined us. Descartes challenges us to see if we can be absolutely sure that we do not actually inhabit such a world!

Modern popular culture has taken up this scenario in various entertaining ways. I think it is fair to say that the worlds imagined in "Total Recall," and "The Matrix" are excellent examples of scenarios that raise the theoretical possibility of false belief that is (at least for those who don't escape the Matrix!) invulnerable to refutation.

Surely the question whether pentagons are more like hexagons than circles just invites the riposte: "more like in what respect?".

If we are interested in whether figures have straight sides and vertices or lack them, then of course pentagons will get put in the same bucket as hexagons, while circles will go in another bucket (with e.g. elipses and parabolas). It's an objective fact that pentagons are like hexagons (and not circles) in having straight sides and vertices.

If we interested in whether we can tile a plane with (regular) figures of a certain kind, then pentagons will be classed with circles (no, you can't tile a plane with those), and hexagons will belong in the other bucket along with e.g. squares and triangles. It's an objective fact that pentagons are like circles (and not hexagons) in that you can't tile a plane with them.

So we might say that the bald question "are pentagons more like hexagons than circles?" is incomplete. It needs to filled out (either explicitly or by context) with some indication of the resemblance-respects which we care about when we ask. But once those indications have been provided, it can (as our examples show) be a straightforwardly objective matter what the answer is.

Similarly, the question "are the paintings of Monet more like those of Renoir than those of Picasso?" again invites the reply "like in what respect(s)?". Where they were painted? Average market value? Average size? Well, it's perfectly objective what the right answer is, if it turns out that it's resemblance in one of those respects which is question. But of course, we might very well be more interested in some more purely aesthetic features; and you might suppose that now things do indeed become more subjective. But the issues don't become more subjective because they are issues about resemblances but because they are issues about the aesthetic. (Another possibility is that the questioner just isn't clear what kinds of resemblance are being asked about: but then we don't have a determinate question to answer.)

In sum: at least some questions about resemblances -- once it is made clear what resemblances are in question -- can have just as objective answers as other questions about a thing's properties. There isn't anything instrinsically subjective about issues of resemblance per se: it depends what kind of resemblances you are interested in.

Absolutely.

There is nothing stopping you from defining an analytic theorem of a formal system to be one whose derivation requires appeal to at least one member of a designated subset of axioms. But on what basis are you deciding to single out that particular subset of axioms? If you say you're being guided by the fact that those particular axioms express truths about meanings, whereas other axioms express substantive truths about the world, then you owe an explanation of what that distinction amounts to -- and arguably, that will be no easier to give than an outright analysis of "analytic". (You might also look at W.V. Quine's discussion of Semantic Postulates in his paper "Two Dogmas of Empiricism.")

I don't know what you mean by: "One can create axioms that make statements like 'all bachelors are married' true". I assume that by 'married' you mean 'unmarried'. But I still don't understand. Perhaps you mean that one can write down obviously true principles from which the truth of every analytic sentence, and no other sentence, follows. But we can't.


Not yet anyway.