# I have a question about Whitehead and Russell "Principia Mathematica". Can mathematics be reduced to formal logic?

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, 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. 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...

# Are mathematical statements existential statements? I ask because we're taught that set theory is, in a sense, foundational to all mathematics, and most of the propositions considered in set theory essentially assert the existence of particular sets.

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 --...

# Two questions. It seems that no one has figured out good standards for acceptance or rejection of philosophical arguments. In science, observation is king. If evidence contradicts a theory under careful conditions, the theory is false. In math, we justify things formally; we cannot expect more certainty. So would you agree that philosophy, as a field that aims at knowledge and not something else like evoking emotions, suffers from a lack of standards? And since at the moment I suspect it does, I want to ask also, why do philosophers act so certain? To them their arguments are true or correct (or whatever) without empirical evidence or rigorous proof. They should be the most uncertain people of all, even more so than scientists. And they are pretty darn humble. (A better way to ask this might be, aren't proof and evidence the two best ways to knowledge? If so, shouldn't philosophers be much more uncertain than they appear (to me)? I now realize it's dependent on how I see things, so I only hope you can...

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...

# Much of philosophy is concerned with providing a rigorous foundation to truths which are otherwise intuitive and uncontroversial; think of philosophy of math, for instance. Do philosophers believe that, absent an appreciation of such foundational principles, laymen don't actually "know" such truths, e.g., that 1+1=2; and if laymen do know such truths, how do they know them?

Actually, the presumption here is wrong. It isn't the case that "much of philosophy is concerned with providinga rigorous foundation to truths which are otherwise intuitive anduncontroversial". In particular, that isn't the case in the philosophy of mathematics. Of course, famously, Frege tried to show that the basic laws of arithmetic (and hence the proposition 1+1 = 2) can be derived from the laws of logic plus definitions. But he did this in order to defend the claim that arithmetical truths are analytic, true in virtue of logic alone, and so explain why those truths are necessarily true and why they necessarily apply to everything. He didn't claim that, prior to his attempted derivation of 1+1 = 2 from pure logic, no one knew it to be true. Rather we weren't in a good position to see clearly the sort of truth that it is, analytic according to Frege. Unfortunately, one of Frege's putative laws of logic turned out to lead to contradiction, and his foundational edifice crumbled (though neo...

# Can philosophy of mathematics influence mathematics, or it is just an abstraction of what actually works?

Three examples to think about. First, Frege's invention of the predicate calculus was driven by philosophical reflection on the nature of quantified propositions, and led in turn to modern mathematical logic. Second, the so-called Hilbert programme was driven in part by more philosophical reflection, this time on the limits of what we can directly "intuit" to be mathematically correct; that programme led in turn to the development of modern proof theory. Third, Kurt Gödel's philosophically driven work on set theory was mathematically hugely important. [Sorry, those reference links are inevitably to material that quickly gets mathematically heavy!]So, it surely is the case that specific philosophical ideas -- philosophical reflections on foundational matters -- have influenced the development of mathematics. And one might say too that a more general set of philosophical ideas about the proper nature of mathematics drove the whole Bourbaki project which has been so influential in the...

# If we prove that a proof exists, why isn't this effectively the same as finding the actual proof?

To start with a story. Once upon a time, I used to teach introductory logic using Lemmon's textbook. And some exam questions would have the form "Use truth-tables to test the following arguments for validity; in the cases where the argument is valid, provide a proof from the premisses to the conclusion in Lemmon's system". Given that Lemmon's natural deduction system is complete, a student who correctly did a truth-table showing that a particular argument is valid thereby proved that there is a proof from the premisses to the conclusion. But of course, she had to do more to answer the second part of the question, and get the marks! She had to give an actual natural deduction proof. This little story reminds us that, quite often, we aren't just looking for any old proof, but for a proof of a certain style S , a proof that uses certain kinds of resources. And proving that an S -proof of some result exists isn't in general to give an S -proof. This sort of point applies outside the...