I don't have expert knowledge about the epistemic status of mathematical truths, but I can report on my intuition that it is perfectly clear that there could exist a society that genuinely believed that it was morally acceptable to torture innocents for pleasure but not at all clear that there could exist a society that genuinely believed that two plus four was equal to five. So, my sense is that the two claims you present have different statuses.

Your question, however, is not about whether a society could treat that statement about the immorality of torture as false. And, likewise, your question is not about whether or not there exist specific moral beliefs or systems that endorse the truth of that statement. Instead, the heart of your question is the thought that there might exist "moral laws" that demonstrate that the statement condemning torture is necessarily true.

About the notion that the universe functions according to moral laws, I would draw your attention to a recent discussion by the social philosopher Charles Taylor about the “moral order" that he believes is dominant in our society. One of the lessons Taylor draws is that societies are influenced by a "moral background" that, in turn, is determined, in part, by underlying "picture of society" or "social image" that can vary over time and across different cultures. If Taylor is right, it wrong to assume that the "universe functions" according to invariant moral laws and so the thought that it does function in that way could not ground the necessary truth of your statement about torture.

(To say a bit more about Taylor's fascinating historical account: He believes that the “modern moral order” centers on the ideas that "Human beings are rational, sociable agents who are meant to collaborate in peace to their mutual benefit" and dominates modern political thought because it conforms to a specific "social imaginary" that developed in our culture contingently and through an extremely complicated historical process. See his A Secular Age, pp. 159 and following.)

I'd like to suggest a rather different take. Your question makes most sense on the assumption that there can be objective moral truths; if there can't, then no universe "functions" in accord with any moral laws. So let's assume, at least for the moment, that there are such things as objective moral truths. And now let's make a bit of a distinction. Let's agree that as things stand, it's wrong to use taser guns on babies. Could there be a universe where it was perfectly acceptable to taser a baby? If we suppose that babies are wired differently in that universe, the answer could well be yes. Perhaps the nervous systems of babies in this distant universe are set up so that applying the taser provides some sort of painless and beneficial stimulation. And so somthing that's wrong in our circumstances would be right in that far-off world, but only because some background non-moral facts differ.

Now it may be that background facts about our social arrangements and our ways of understanding our own actions are among the background facts that make a moral difference. A trivial example: in a culture where belching is a way to compliment one's host, it's just fine to belch after a good meal. It's a merely contingent fact that as our society developed (or mine, in any case), belching at the dinner table would be a way to insult one's host. What we actually happen to care about, value and abhor is a complicated matter of fact, and on the view I'm interested in, non-moral facts of all sorts can be relevant to right and wrong.

That said, however, could there be two universes that were alike in all these non-moral ways and yet differed on what's right and wrong? Could the moral facts vary independently of all else? I'll confess that I can't see how. And so I'd say: if there are objective moral facts, the most plausible view is that they "supervene" on other sorts of facts. What that means is that there can't be a difference at the level of moral truth without there being a difference in the non-moral facts. Putting it another way, once all the non-moral facts are fixed, it's plausible that all the moral facts are fixed as well.

Of course, if there are no objective moral facts, none of this is so. But then, if there are no objective moral facts, the answer to your question is no because the universe doesn't function according to any moral laws in the first place.

A footnote, however: I'm a little uncomfortable with the phrase "function according to moral laws," because it suggests that moral truths are like physical laws, and that seems to me not to be the right picture. But that's another story, and a long one at that.

A philosopher of mathematics is interested in questions like: what are numbers? what kind of necessity to arithmetical truths have? how do we know the basic laws of arithmetic are true? what about sets -- do they really exist over an abover their members? is there a universe of sets? there are various set theories, how can we decide which is true of the universe of sets? And so on. You don't have to be a working mathematian to think about these matters (though obviously you have know a little about the relevant mathematics you want to philosophize about). Nor do you have to be a logic-expert.

Most mathematicians aren't interested in the philosophy of mathematics (just as most scientists aren't interested in the philosophy of science, and most lawyers aren't interested in the philosophy of law). They just go about doing their maths. And among mathematicians, a serious interest in logic is a rarity: you can certainly be a mathematician without being a logician (e.g. by being a fluid-dynamicist or an algebraic topologist). Of course, mathematicians use logical reasoning. But you can be a mathematician without being concerned with the articulation of a formal theory about good reasoning.

And I suppose you can at some level be a logician of sorts without being a serious mathematian. People write whole books on `informal logic', without being mathematicians (or being interested in the philosophy of mathematics).

Kurt Gödel though was indeed not just a great mathematician, who was concerned with formal theories of logic (so was a mathematical logician in particular), but he was also profoundly interested in issues the philosophy of logic and mathematics.

Indeed, Gödel thought that his philosophical ideas were in part responsible for some of his major logical discoveries. For example, there was a tendency in the 1920s to run together the idea of a proposition's being a mathematical truth with the idea of its being provable. Gödel thought this was a philosophical confusion. He was a platonist, in the sense of holding mathematical truth to be one thing, and our being able to get at the truth via a proof is something else. And clearly distinguishing truth from proof enabled him to discover his First Incompleteness Theorem which shows that, for any 'nice enough' theory containing a bit of arithmetic, there will be truths of arithmetic it can't prove (for a bit more explanation of that, the reasonably accessible first chapter of my book about Gödel's incompleteness theorems is freely available here!)

This question raises a number of issues it is worth disentangling.

1. It is far from clear that we should think of our reasoning as "operating on a set of axiomatic beliefs". That makes it sound as if there's a foundational level of beliefs (which are kept fixed as "axioms"), and then our other beliefs are all inferentially derived from those axioms. There are all sorts of problems with this kind of crude foundationalist epistemology, and it is pretty much agreed on all sides that -- to say the least -- we can't just take it for granted. Maybe, to use an image of Quine's, we should instead think of our web of belief not like a pyramid built up from foundations, but like an inter-connected net, with beliefs linked to each other and our more "observational" beliefs towards the edges of the net linked to perceptual experiences, but with no beliefs held immune from revision as "axioms". Sometimes we revise our more theoretical beliefs in the light of newly acquired, more observationally-based, beliefs: but also sometimes we revise some observational beliefs (e.g. diagnosing perceptual error) on the basis of well-confirmed theory. It's not a one-way process, always reasoning from "axioms" to our remaining beliefs, but a complex balancing act, trading off considerations of theoretical organization against the observational appearances to keep the net both useful and coherent.
   
2. The notions of consistency and validity are indeed connected. Roughly, an inference is logically valid just if the premisses taken together are not consistent with the negation of the conclusion. But note that this connection ties establishing validity with establishing inconsistency. And establishing inconsistency is easy compared with establishing consistency. For to establish an inconsistency, we just need to find one compelling argument that leads to a contradiction: while to establish consistency, we have to show that, take any argument you like, it won't lead to contradiction. That means that, even if we are not sure that our repertoire of logical apparatus is overall consistent (we are not sure that there's no lurking contradictions anywhere), we can still be sure that one particular argument is valid (i.e. sure that one particular collection of propositions does lead to contradiction). That's why, contrary to the implication in the question, doubts about consistency claims don't automatically spread into doubts about validity claims.
3. But set those two points aside. There remains the following issue (which I take to be the key intended question). Suppose that we regiment our principles of logical reasoning into One Big Theory. Suppose we worry whether the One Big Theory is consistent. Then how can we show that the One Big Theory is in good order after all, except by using reasoning of the kind embodied in the One Big Theory whose status is in question? It looks as if we are doomed to go round in circles. To make this more pointed: it is arguable that our logical/mathematical reasoning can be regimented inside a set theory like ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice). What if we worried whether ZFC is consistent? Then using ZFC to argue that it is consistent would hardly be convincing. Indeed it's worse that: in fact, if ZFC "proved" its own consistency, not only would that not convince a skeptic of its consistency (for an inconsistent theory would "prove" anything, including its own consistency), but it would actually demonstrate the theory's inconsistency, by Gödel's second incompleteness theorem. So doesn't this show that we shouldn't trust ZFC?
   
4. No! For note, we just haven't any good reason to suppose it is inconsistent. It isn't that we have grounded worries about ZFC's consistency and so we are casting around for some independent proof to quiet those doubts. So we don't need a "consistency proof". Decades of intensive study have turned up no inconsistencies; and that's no surprise, as ZFC is custom-designed to avoid the problems that lead to inconsistencies in more "naive" set theories. The theory is in great shape, and by this stage in the game it seems to be mere idle skepticism to imagine things might be otherwise. And while idle skepticisms might be fun for half-an-hour in a philosophy classroom (though the game soon palls), they are no reason to stop trusting our mathematics!
   

No, mathematicians haven't defined any meaning for expressions like "infinity-th" or "infinity+1-th". (The fact that they're so awkward to write should be something of a giveaway!). It's important to appreciate that infinity is not a number. Don't be misled by the fact that we can say things like "there are infinitely many natural numbers", which seems to have the same form as a sentence like "there are three coins in the fountain". The number sequence doesn't go: 0, 1, 2, 3,... 1,000,000, 1,000,001, 1,000,002, 1,000,003,... infinity, infinity+1, infinity+2, infinity+3.... Rather, infinity is a property of certain sets, such as that of the natural numbers. The infinity of that set consists in the fact that, for any member you might care to consider, there will be another member, which is larger than it but which is nevertheless still finite. And we can easily refer to that set as a whole, and we can even quantify universally over its members. We can say, for instance, that they all have successors, or that this set contains them all. (Though I'm not sure just how distinct this is from saying that each of them has a successor, or that each of them is a member of the set).

Given that infinity+1 is not a number, there is no such thing as an infinity+1-th position in a hypothetical enumeration of the real numbers, where you could add an extra one from the diagonalisation of the sequence. But that's no problem, as there's a very simple alternative: just shift all the members up one place, and stick this extra one at the start. But then you'll have a new sequence, and it too will be susceptible to diagonalisation. The diagonally-constructed number you get for this new sequence won't be the same as the one you got the first time (it will differ from that one in at least its first digit): but there will still be one, and it's this that shows that the real numbers are uncountable. The uncountability of a set means that any list, with a first member, a second member, a third member and so forth, will fail to contain all of them. Try to construct such a list of the real numbers, and you'll invariably find that you've missed one out, namely the one you get from diagonalisation.

But the natural numbers are countable. You can place them in a list where the first member is zero, the second is one, the third is two, the fourth is three, etc. (It's tidier with the positive integers!). Now, if you stop after only the first n natural numbers, then, yes, the number m that you've defined will indeed be absent from your list. But this fact doesn't carry over to the infinite case. If you take the whole sequence of natural numbers and add them all together, they won't give you a natural number as a result, whereas the diagonalisation of a purported enumeration of the real numbers will give you a further real number -- that's the difference between the two cases. Mathematicians are comfortable with the notion of sums to infinity of geometrical progressions, but only when these yield finite results. Thus, for instance, the sequence 0.5, 0.25, 0.125, 0.0625... issues a sum to infinity of 1. But there is no sum to infinity defined for the arithmetical progression of 0, 1, 2, 3,.... And, if there should be some temptation to say that such a sum does indeed exist, and it's infinity, just remember again that infinity is not a number, and still less is it a natural number. Consequently, it is not the case there "there is always a natural number m that is not counted in the set", and the set of natural numbers can comfortably remain countable after all.

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.

Let's narrow the question a bit: can arithmetic be reduced to logic? If arithmetic can't be so reduced, then certainly mathematics more generally can't be.

What would count as giving a reduction of arithmetic to logic? Well,

1. We would need to give explicit definitions (or perhaps some other kind of bridging principles) relating the concepts of arithmetic to logical concepts. Otherwise we won't get arithmetical concepts into the picture at all.
   
2. We would need to show how the theorems of arithmetic can in fact be derived from logical axioms plus those definitions or bridge principles.

But what kind of definitions in terms of what sorts of concepts are we allowed at step (1)? And what kinds of logical principle can we draw on at step (2)?

Suppose you think that the notion of a set is a logical notion. And suppose that you define zero to be the empty set, one to be the set of all singleton sets, two to be the set of all pairs, and so on. We can then define the successor, addition and multiplication functions in reasonably neat ways, and so get translations of arithmetical claims into claims about sets. Now suppose you also allow some basic principles about sets to count as logical axioms. Then, lo and behold, you'll be able to derive arithmetical truths from the definitions by using 'logic' (a distant echo here of how Whitehead and Russell in fact proceeded).

Job done? Well, most of us nowadays would say "no", because we wouldn't take the notion of a set to be a purely logical one, or set theory to be logic. But Frege thought otherwise, taking the notion of the extension of a predicate to be a logical notion. And although Russell traded in extensions for "propositional functions", he did think that a rich theory of such entities counts as logic. This isn't the place to argue for the modern narrower view of the scope of logic as against Frege or Russell. But mentioning this divergence highlights one key issue involved in answering the initial question: how much "logic" are we allowed?

With hindsight, we can see that well before Principia, Frege got to what now seems a really key result. Suppose you take the bridge principle (unhistorically called "Hume's Principle")

The number of Fs = the number of Gs if and only if there is a one-one correspondence between the Fs and the Gs.

Then this time define zero to the the number of things that aren't self-identical, one to be the number of things identical to zero, two to be the number of things identical to either zero or one, and so on. We can again add definitions of successor, addition and multiplication functions in reasonably neat ways. And then -- and this is "Frege's Theorem" -- the bridge principle plus those definitions plus second-order logic gives us arithmetic.

Now even if we balk at counting set theory as logic, maybe second-order logic is properly so-called (though actually, quite a few will dissent from this, following Quine in counting second-order logic as actually just some set theory in sheep's clothing -- but let that pass.) However, we still have to ask: is Hume's Principle an acceptable sort of principle to use in a reductive endeavour? It seems that its left hand side introduces entities (numbers) which don't appear on the right-hand side (which just talks of their being a suitable one-one function with domain the Fs and range the Gs). So Hume's Principle, many will say, can't be a genuine definition as it imports new ontology. Hence Frege's Theorem doesn't really reduce arithmetic to something else, i.e. logic plus definitions. At best, though still interestingly, it reduces arithmetic to just one core arithmetical principle (plus logic and definitions). But again, this isn't the place to argue about the status of Hume's Principle. But mentioning the question now highlights another issue involved in answering the initial question: what kind of bridge-principles linking arithmetic to logic are we allowed?

If we are a bit stern about what counts as logic, and stern about the kinds of bridge principles that are allowed in a successful reduction, then no, arithmetic surely can't be reduced to logic. But some (especially so-called neo-logicists) do think we can be less stern and rescue something true and important from the old attempts to reduce arithmetic to logic.

First, we need to set an issue aside. The word "induction" is sometimes used to refer to a certain sort of mathematical argument in which we prove something for every case by showing it for a "base" case and then showing that if it holds in the first n cases, it holds in the n+1th case. But it's pretty clear that your question is about induction as a matter of reasoning empirically from a limited set of instances to a claim about all cases, and so we'll use the word "inductive" in that way below.

Here's an example of a necessary truth: every star that has planets orbiting around it is a star. Notice that it's universal; it says something about every star with planets. And if it were like "every star with planets orbiting around it is at least 3 billion years old," we could only show that it was true (assuming it is) by empirical means and thus, in a loose sense, inductively. (What we'd actually do is produce an argument from various theoretical and observational premises, but at some point, something like induction would no doubt enter the picture.) In fact, however, our knowledge that all stars with planets are stars is secure without the need for inductive reasoning. That's because any proposition of the form "Every x that is a both P and a Q is a P" is a truth of logic.

Just what logical knowledge amounts to and how best to account for our having it are big questions. But it seems pretty clear that inductive reasoning presupposes deductive reasoning. Imagine some principle of inductive reasoning that you want to invoke in coming to an empirical conclusion. The principle will be a generalization: it will say somethng about all cases that fit a certain pattern. To get to the case at hand, you have to be able to make the deductive move of applying the generalization to the particular case. We could elaborate this point quite a bit further. (For some amusing related reflections, read Lewis Carroll's "What the Tortoise Said to Achilles.")

As for axioms in mathematics, there's a sense in which communities agree to them, but it's not clear that this takes away from their objectivity. Consider a set of axioms for Euclidean geometry. We agree amongst ourselves that these axioms fix what we call Eucliden geomety whether or not any actual spaces are Euclidean. The Pythagorean theorem follows from those axioms whether or not there are any Euclidean right triangles. And so what we know is this: if there are any situations where the axioms of Euclidean geometry are true, then any right triangles in those situations have hypotenuses whose squared lengths are the sums of the squared lengths of the sides. We don't need to pursue any tricky empirical investigations to be confident of this.

None of this is meant to say that all mathematical truth can be reduced to logical truth. And it may even be true that we can have inductive reasons for being confident of some purely mathematical claims. (Some computer scientists talk about "empirical mathematics.") But the mere fact that a mathematical proposition says something about an infinite set of cases doesn't show that our knowledge of it is empirical or inductive.

I'd separate the question whether mathematical statements are (often) existential from the question of the status of set theory. (Sure, we can construct faithful proxies inside set theory for most of the structures that mathematicians are interested in. But it is a moot question whether this makes set theory foundational in any good sense at all.)

Now, many mathematical statements are pretty uncontroversially not existential, but have the form "if anything is A it is B". So the theorem that anything which is a finite division ring is commutative doesn't tell us that there are such things as finite division rings, but only what they must be like if they do exist.

But of course many other common or garden mathematical theorems certainly do look existential. "There are an infinite number of prime numbers" looks existential -- and it is naturally read as implying that there are prime numbers (lots of them!). "There are four regular star polyhedra" looks existential -- and it is naturally read as implying that there are regular star polyhedra. And so it goes.

Perhaps appearances are deceptive, however. Perhaps these superficially existential statements are really non-committal "if ... then ..." statements in disguise. So really what we are saying, for example, is something along the following lines: if any collection of things has the natural-number structure, then it contains an infinite number of things filling the prime-number role -- leaving it open whether there is anything that satisfies the antecedent of the conditional. Alternatively, maybe "there are an infinite number of prime numbers" is a statement made inside an essentially fictional mode of discourse (the arithmetical fiction), and no more really implies that there are numbers than "Sherlock Holmes lived at 222B Baker Street" implies that there really was such-and-such a man living along Baker Street.

Now, "if ... then ..."-ism and fictionalism are problematic as stories about the status of mathematics. I'm not recommending either position! But they are enough to illustrate the point that it isn't, perhaps, so obvious after all that prima facie existential mathematical statements have to be construed as really being kosher existential statements.

Stewart Shapiro's Thinking about Mathematics gives a terrific introduction to some of the issues hereabouts.

The kinds of reasons that are given for favoring one scientific theory over its rivals are a good deal more subtle than "observation is king." To begin with, a theory need not be justly rejected merely because it conflicts with a given observation; sometimes, the observation is appropriately doubted, and sometimes, a given theory is rationally retained despite its failure to fit our observations because blame for the mismatch is placed on other theories ("auxiliary hypotheses") that were used to bring the theory to bear on those observations. (The Copernican model of the solar system, for instance, was retained despite 300 years of failure to observe the stellar parallax it apparently predicts.) By the same token, a theory that fits our observations very well may nevertheless be justly and emphatically rejected on the grounds that it is ad hoc, fails to fit nicely with our other theories, lacks unity or fruitfulness or explanatory power, etc.

Once these familiar features of scientific practice are recognized, then I think the choice among theories in philosophy seems not so dissimilar to the choice among competing scientific theories. Admittedly, much of philosophy is a priori, unlike science. However, the virtues of elegance, parsimony, unity, coherence, explanatory power, and so forth play significant roles in both scientific and philosophical theory-choice.

Finally, philosophy has long been and is increasingly brought into contact with empirical results. A philosophical analysis of causal relations, for instance, that fails to do justice to modern physics has a severe deficiency. Of course, philosophers will disagree about the extent to which modern physics (or even classical physics) discovers causal relations, as well as disagreeing about what physics has revealed about them. Nevertheless, philosophical theorizing hardly takes place in isolation from empirical results. Recent literature in the philosophy of mind and perception offers a host of tremendously potent examples of this.

Just a footnote to Marc Lange's response (which seems spot on to me). It is worth adding that in serious analytical philosophy there is actually a good deal more agreement on arguments than there might appear to be at first sight, and there is a good deal of pretty secure knowledge.

For what often emerges from the to and fro of debate is essentially something of the form "If you accept A, B and C, then you'd better accept D too". Then one party might endorse A, B and C and conclude that D; and another party might think D is unacceptable, and conclude that one of A, B, or C must be wrong. And another party again (me, often!) might not know how to respond. [A trite example. If you accept act utilitarianism plus some other things, it seems that you should sanction the sheriff hanging an innocent man if that is the way to stop a riot in which more innocent people are killed. Some bite the bullett, some think so much the worse for utilitarianism.]

Now, there may indeed be a loud disagreement between the first two parties. But a lot of hard thinking may have gone into working out what they agree on, namely that "If you accept A, B and C, then you'd better accept D too". [For example, initially people might have thought, wrongly, that accepting A and B alone forced conclusion D, and it took some subtle argument to show that C was playing a role too. Working out what act utilitarianism really does sanction is like this.] And finding that sort of connection can represent a solid achievement of which we can be tolerably certain, even while there remains vigorous disagreement about what to do with the discovery.