Episode post here. Many thanks to David North for this transcription.
Matt Teichman:
Hello, and welcome to Elucidations, a philosophy podcast
recorded at the University of Chicago. I’m Matt Teichman.
Jaime Edwards:
And I’m Jaime Edwards.
Matt Teichman:
With us today is Patricia
Blanchette,
Professor of Philosophy at the University of Notre
Dame. She is here to speak with us about the
logicism of Gottlob
Frege. Patricia Blanchette, welcome.
Patricia Blanchette:
Thanks very much, Matt, Jaime.
Matt Teichman:
Logicism is usually described as the research program in
philosophy from the end of the 19th/beginning of the 20th century, whose basic
idea was that mathematics could be reduced to logic. What does that mean
exactly? What would it be to explain mathematics in terms of logic, or reduce
mathematics to logic?
Patricia Blanchette:
Well, it turns out there are a number of different
things people have meant by that. What Frege meant by it was pretty
straightforward. He thought that most of mathematics, which includes
arithmetic—that is, the theory of your ordinary numbers, 1, 2, 3, 4, 5, plus
the rest of the integers, the negative numbers, plus the rational numbers, and
the real numbers—all of those, but not including geometry. So what I’ll use
the term ‘arithmetic’ for covers all the stuff having to do with all the
numbers.
Frege thought that the truths of arithmetic, things like ‘two plus two equals four’, ‘every prime number has a next prime number after it’, and things like that—all of those facts about the numbers could be proven using just principles of logic. So, the reduction, for him, was a matter of proof. You can prove every arithmetical truth just using logical principles.
Jaime Edwards:
So the claim is that all mathematical truths are
logical. This is not a claim about how mathematical truths are discovered or
applied. Is that correct?
Patricia Blanchette:
That’s right. Frege really didn’t take himself to
have a psychological thesis. He didn’t take himself to have a thesis about how
children actually learn the arithmetic that they learn. But he did take his
logicist claim to be important for understanding something about the foundations
of our overall knowledge of arithmetic. So there’s a sense in which he thought
that because arithmetic really is just complicated logic, you can explain how
people know arithmetic just by explaining how they were in a position to know
principles of logic.
And that, for him, meant that you can because you can know principles of logic just by pure reasoning—just by thinking hard, and by thinking about things like this: ‘well, if some claim is true, then the denial of that claim is false’. That’s a principle of logic. Or if you have a claim, call it P, and a claim Q, if the claim ‘P and Q’ is true, then P is true. To know that, all you have to do is think about what ‘and’ means, and reason a little bit.
So that’s what he means by logical knowledge—that kind of pure reasoning. So, for him, the claim that arithmetic is reducible to logic implied that people can in principle come to know truths of arithmetic just by doing that kind of reasoning, in a somewhat complicated way.
Jaime Edwards:
So the motivation for the program of logicism, then, is
something like: giving an account of the nature of mathematical truths. Is that
the motivation?
Patricia Blanchette:
Yes, absolutely. Frege thought that he was going
to give an account of the nature of mathematical truth, and therefore of
mathematical knowledge. Yes.
So in particular, he disagreed with people who thought, for example, that mathematical truths are about collections of objects in the world, in such a way that you would learn the truths of mathematics by manipulating pebbles, or something like that. This was a view that he attributed—maybe accurately, maybe not—to John Stuart Mill.
Mill, who was an empiricist as you know (someone who thinks all of our knowledge comes through our senses), thought: how do you learn truths of mathematics? Well, you look at two pebbles, and you look at three pebbles, and you push them together in a pile, and you count five pebbles. You do that a few times, and then you come to know that 2 + 3 = 5. This is a theory that—it’s not as absurd as I’ve just made it sound, and there may be something to it—but it’s a theory that’s very difficult to use in order to explain how we know things about very large numbers, or how we know things about irrational numbers, or how we know things about infinite collections of numbers. So, ‘two plus three equals five’ is easy, but most of mathematical knowledge is hard to explain in terms of our empirical access to piles of pebbles, or anything that we can know about through our senses.
Frege thought that was the wrong way to go. And he also thought that various other accounts were wrong about the nature of mathematical knowledge. He thought that once you understand that mathematics really is just complicated logic, you will then know we don’t need our senses to come to learn the truths of mathematics. And we don’t need anything else other than our ability to reason in purely logical ways. That was the idea.
Matt Teichman:
So, it’s very intuitive that if the sentence ‘P and Q’
is true, then the sentence P has to be true. So, P just follows from ‘P and Q’.
Patricia Blanchette:
That’s right.
Matt Teichman:
You know, if I’m wearing a black hat and a red shirt,
then it follows from that that I’m wearing a black hat. But intuitively, this
seems pretty different from the claim ‘two plus two equals four’. So how exactly
do you get from these basic sorts of examples that we’ve been discussing, of
drawing logical inferences, to numbers? How do you translate these statements
like ‘two plus two equals four’ into statements like ‘Matt has a black hat and
Matt has a red shirt’, therefore ‘Matt has a black hat’?
Patricia Blanchette:
That is a really good question, and the answer is
a little bit long and complicated, if we go into the details. But I can sketch
the basic picture.
What Frege thought was that you can take a sentence like ‘two plus two equals four’ and spell out what it means—and I’ll come back to what he thought it means in a minute—spell out what it means in a way that makes it turn out to be somewhat more complicated in structure than ‘two plus two equals four’. Then once you’ve done that, you can give a proof of that, as it were, fully fleshed out, fully analyzed version of ‘two plus two equals four’. And that proof that you give is going to have as its premises only principles of pure logic. That may sound like magic, that somehow you would use principles like the one you gave about the black hat and your red shirt, and get from that to ‘two plus two equals four’. So, let me say a little bit about how the transition goes.
First, let’s start with Frege’s view about numbers are four. He thinks that the ordinary numbers—one, two, three, four, five—are used primarily for counting things. So I’ll say ‘there are three people sitting at this table’. That’s the primary, basic, most important, use of the number three: the number three is something that you talk about when you’re counting things. Let’s start with what it would mean to say there are zero people at this table—because zero is a really nice number, as far as Frege is concerned. To say there are zero people at the table is just to say there’s nobody at the table.
What is it to say there’s one person at the table? Well, it’s to say there’s something, call it x, at the table. And for everything in the world, if it’s at the table, then it’s x. It’s convoluted, but it’s a way of saying that there is exactly one thing is at the table that doesn’t mention the word ‘one’.
How about two? We’ll go as far as two. What does it mean to say there’s two people at the table? It means: there’s an x at the table and there’s a y at the table. And for anything, if it’s at the table, then it is either our x or our y. That means there aren’t any other things. That was a very longwinded way of saying there are exactly two things at the table. You can go on like this and say there are exactly three things, there are exactly four things at the table, without mentioning the words ‘three’ and ‘four’. There’s the beginning of the reduction.
Now, for Frege, what you do is you analyze what numbers are by analyzing those kinds of statements about numbers. The kinds of statements he was interested in were: ‘there are zero such-and-suches’, ‘there’s one such-and-such’, ‘there are two such-and-suches’. And, at the bottom of this, was also an account of claims like this: there are just as many such-and-suches as there are so-and-sos. There are just as many people at this table as there are chairs at this table. And for Frege, what that meant is: you could match up the people and the chairs, so that every person gets a chair, every chair gets a person, nothing is left out, and you don’t have two people in one chair, or one person on two chairs. You just match them up, as we say, one-to-one.
That notion of matching one-to-one turned out to be very, very important for him when he was assessing statements that have to do with numbers. So to say there are just as many of these as there are of those is to say you can match the ‘these’ with the ‘those’ one-to-one. And that means the number of ‘these’ is identical to the number of ‘those’. So, he thought that statements about numbers that say ‘this number is equal to that number’ were very, very, very important, and that you could explain the meaning of those in terms of the matching up one-to-one of the things, on the one hand, with the other things, on the other hand. Like the chairs at the table and the people at the table. That gives you a rough idea of some of what goes on here. He takes talk about ‘three’ and ‘four’ and ‘five’—and talk about prime numbers, and things like that—and explains them in terms of talk that doesn’t mention those numbers themselves, but which, in some sense, he thinks means the same thing.
Matt Teichman:
Yeah, it’s a pretty clever trick, actually, when you
think about it. If you want to define what a number is, you don’t want to just
have the word ‘number’ in your definition of ‘number’. That seems kind of
circular and pointless. You don’t want to say ‘there are just as many people as
there are chairs’ just means ‘well, if there are three people, and there are
three chairs, then the two numbers are the same’. You want to find a way of
defining number without using the word ‘number’ in your definition.
And the way you do that is: you say that there’s a correspondence. A correspondence isn’t the same as a number. It just means: look—for each thing in the one group, I can draw a line from it to something in the other group. That’s the trick that allows us to understand what numbers are in terms of something else: namely, matching pairs of things together.
Patricia Blanchette:
Yeah.
Matt Teichman:
One of the interesting things about translating
mathematical statements into logical statements is that, in principle, it gives
you a new way to check whether the mathematical statement you’re thinking about
is true. Obviously, everybody knows that ‘two plus two equals four’ is true. But
maybe a more complicated one, like a really complicated equation—it’s not
totally obvious whether it’s true or false. Maybe this method of translating
math into logical statements can give us a different way to check to see if the
mathematical claim we’re evaluating is true or false. Was this the idea behind
the logicist program—that you could come up with this automatic way of
checking whether mathematical proofs were correct?
Patricia Blanchette:
No, but that’s a great question. Because when you
look at the work of a logicist—someone like Frege, or some later
logicists—what you see are a bunch of proofs. And they’re very, very, very
careful proofs. They’re much, much, much more careful than anything that gets
published in a mathematics journal, and I can return to the reason for that in
just a moment.
This might lead you to believe that what they’re trying to do is give proofs that are better in the sense of, somehow, ‘more error-free’ than the proofs that a mathematician might give. So you might get the impression that the logicists were trying to make sure mathematicians aren’t really making mistakes.
But that, in fact, isn’t true. Frege, in particular, had no doubts about mathematicians’ ability to prove things. And he had no doubts about the truth of the kinds of things he was setting out to prove—things like, as you say, ‘two plus two equals four’. His reason for giving proofs, and for giving stunningly rigorous proofs, was so that he could be very sure exactly what principles were needed in order to prove the conclusions. The proofs, then, were laid out in a very rigorous step-by-step fashion, not so as to ensure that the conclusions were true—nobody doubted the conclusions—but so as to make it really obvious what principles you needed to use in order to prove them. So that we could see what, as Frege puts it metaphorically, what the truths rest on—what their grounds are, or what it is that guarantees that they’re true.
So his proofs were stunningly rigorous, and one of the things that was neat about his way of proving things was: in order to bring a sense of rigor to the proofs, he invented a new system for doing logic. He invented a new way of writing down proofs such that the steps were much, much, much more rigorous than anything that had been seen before. And even the very formulas that he used to express things like the fundamental principles of logic, or the more complicated truths of artihmetic—the formulas that he used to express them were much better at showing the structure of the truths in question.
And so, what you end up with in Frege is the first system of what we now think of as formal logic. Instead of using a natural language like English or German—which would have been his natural language—instead of using German to write the proofs, he wanted to make sure that he was as explicit and rigorous as possible. So he moved to what we call a ‘formal language’, which is really, really rigorous, and you have to learn it in order to do the proofs. It’s not hard to learn. It’s not like learning a natural language; it only has a few kinds of words in it. But it’s a language that’s really, really good for doing logic in. And it’s a language that evolved into the languages that we now use for doing logic.
So, though Frege’s main goal wasn’t to come up with a system of formal logic—he was doing that just a sideline, just as a means of doing what he wanted to do, which was to prove the truth of logicism—on the way, he did this most amazing thing. He came up with what we call ‘quantified logic’, a formal system which is very, very similar to what we use today.
Jaime Edwards:
So if Frege’s logicism were to succeed, he would solve a
big problem—which is understanding the nature of mathematics—because he
would be reducing it to logic, and thereby understanding it in logical
terms. Does he think that understanding the nature of logic is easier to handle,
or is that itself going to be a major task?
Patricia Blanchette:
Nice question. That itself is, in some sense, a
big task. It’s not one that he took himself to have very much to say about. So
you might wonder why it is that showing that mathematical knowledge really is
just a species of logical knowledge—why does that help? After all, it might be
kind of hard to understand what logical knowledge is all about. That’s a very
good question. And one thing you might say, at the very least, is that it would
reduce two hard questions to one hard question, which, perhaps, is progress. But
in fact, it may be a bit better than that, and I think Frege thought it would
have been a bit better than that.
So let me back up and say why mathematical knowledge is so interesting and hard to explain. When we think about the different kinds of knowledge that people have, it looks like we can split them up into two camps. One camp is the kind of knowledge that you need to use your senses to figure out. Mundane things like ‘the sun is shining today’, but even more general things like the basic principles of chemistry and geology. We know those by experiment. We use our senses. We have what we philosophers call ‘empirical data’. So those parts of our human knowledge must be explained by looking at the way in which we get information in through the senses. And then, we combine that, and we get general truths about the world, and indeed, truths about the world that we sense. So, that’s how chemistry, geology, physics, astronomy, all of those sciences work.
Mathematics looks very different. If you go to the mathematics department, you don’t find any microscopes, you don’t find any telescopes, you just don’t find anything that makes it look like mathematicians have to use their senses very much. What mathematicians do, it seems, is they just sit around and think. And they can do it in the dark if they want. They just don’t need anything that comes in through the senses. But that’s the cool thing about mathematics: you don’t need information about the empirical—that is, sensory. You don’t need to know anything about the empirical world in order to come to know the truths of mathematics. But you need mathematics in order to do anything in the empirical world. You cannot build a bridge without knowing your calculus; it’ll fall down.
So that’s a really, amazingly cool thing, I think, about mathematical knowledge. It looks like it comes in a non-empirical way, but in order to do anything out there in the world, with the empirical world, you need mathematics. And so, the question is: how can mathematics be so useful about the world of our sensation, but not be grounded in the world of our sensation?
Let me just put this in a contrast. Suppose you wanted to do a chemistry experiment. Well, you’d have to know how the chemicals are going to interact. But in order to figure that out, you have to have experience of chemicals. And so, it makes sense that the chemical knowledge you have comes from experience, and it’s useful for experience. But math looks weirdly different. It doesn’t seem to come from experience, but it is useful for experience.
And so, this puts us in a position of thinking of mathematics as uniquely interesting to explain. That is to say, our mathematical knowledge looks particularly uniquely interesting to explain. So now, the thing that would come out of reducing mathematics to logic might be this: you might discover that ‘oh I see why mathematics is so useful—logic is pretty darn useful’. If I want to know that it’s Tuesday, perhaps all I need to know is there are only seven days of the week and it’s not Monday, and it’s not Wednesday, and it’s not Thursday, and it’s not Friday, and it’s not Saturday, and it’s not Sunday. It’ll follow from that that it’s Tuesday.
And I can use logic to figure things out about the empirical world, sometimes. If I know that all snails have this particular physical feature, and that’s a snail, I can deduce—without even looking at it—that that snail has this particular physical feature. So, logic, it looks like, actually has the characteristic we’re interested in. Logic is useful in the empirical world. We use it all the time; you can’t do anything without using logic. But, in order to know the truths of logic, you don’t have to have any empirical data. And so, it might be that that is still mysterious—and indeed, I think, in some sense it is—but it doesn’t look mysterious that I can reason from ‘all snails have this feature’ and ‘Snuffly is a snail’ to ‘Snuffley has this feature’. There’s nothing magical about that, it’s just what I mean by these words.
So the fact that logic has these characteristics—that is to say, logical knowledge is not empirical but it’s very applied; it’s very useful—means that if I can show you that mathematical knowledge is just a species of logical knowledge, I will have solved a large part of the really interesting problem of how mathematical knowledge can be like that. That’s one of the reasons that logicism is a really attractive, interesting thesis, and it would have been nice if it had turned out to be true.
Matt Teichman:
So, we just said that it would have been nice if
logicism had turned out to be true. That suggests that there was some sort of
deep problem with logicism. And indeed, the philosopher Bertrand Russell
discovered that Frege’s logicist theory led to a certain paradox. This led to
what you might call a collapse in, at least, the original version of
logicism—a collapse in the whole enterprise. So what happened there?
Patricia Blanchette:
Well, great question, although let me correct one
small thing. What Russell revealed to Frege was not that Russell’s logicist
thesis—namely, the thesis that arithmetic is reducible to logic—he didn’t
reveal that that leads to a contradiction, or a paradox. What he revealed was
slightly different: that Frege’s way of trying to prove logicism was
problematic. And so, let me say something about what the problem was.
First of all, just as background, it’s probably worth knowing that logicism really was a huge project for Frege. It was really, I think, the focus of his life’s work, though he did a lot of other interesting things, which maybe we’ll get a chance to talk about. Really, logicism was what he was trying to do throughout most of his life. He was trying to demonstrate, in an unbelievably rigorous way, that truths of arithmetic are truths of logic. He did this in a series of books. The first one—not really a book, just sort of a monograph, in 1879—then something else in 1884 that pushed the project further, and then another book that came out in two volumes, the first volume in 1893 and the second volume in 1903. It was actually at the printer, apparently, in 1902 when Frege got a letter from Russell.
Let me say something about this book that was at the printer in 1902. It was supposed to be the second of probably three volumes that were absolutely conclusively demonstrating the truth of logicism, by giving these unbelievably rigorous proofs. These are proofs that it’s really hard to even work through them, let alone to come up with them. They were the fruit of years and years of dedicated hard labor, on Frege’s part. And again, the labor was a matter of taking a bunch of very, very primitive—what looked like really basic, obviously true axioms of logic, and using them to go step, by step, by careful step and getting, at the end, a truth of arithmetic. And so, that was the project.
Anyway, in 1902, in the summer, Frege got a letter from Bertrand Russell, who had been working through Frege’s book. And Russell said, ‘I think there’s a problem here. I think that given what you think of as an axiom of logic—something that should only lead to truths—I think you can prove a contradiction’. That is, you can prove something false. And Russell was absolutely right. There was a fundamental problem at the very heart of Frege’s logical system.
The one axiom is called Axiom 5, and it’s an axiom that tells you about certain kinds of objects which Frege called ‘value ranges’, but they’re very, very similar to what we today call ‘sets’. So I will talk about them as if they’re really sets. (There are some problems with doing that, but in any case, we’ll call them sets.) This principle was really important to Frege’s system because the numbers, on Frege’s way of analyzing arithmetic, were supposed to be sets. And so, when Frege’s principle—that basically tells you what kinds of sets there are, and what the basic truths about them are—when that principle turned out to be false, that meant that the whole way of proving truths about arithmetic turned out not to work. That is, again, one of the principles that Frege thought was logical turned out not to be logical, because it led to falsehoods. And it was a principle that was really, really, really important.
And it turns out that, at least as Frege understood it, there really was no way to resuscitate the program. You couldn’t just make a little fix here or there. Frege thought that Russell’s paradox—which is the paradox that you can deduce using Frege’s system—Frege thought that Russell’s paradox showed that his whole approach to the analysis of arithmetic and the proof of arithmetical truths was just bad. It was just problematic from the beginning.
Now, there are some people these days, in the 21st century, who think that Frege was a bit hasty in dismissing the logicist project in light of Russell’s paradox. And so there’s a new movement that’s called neo-logicism, and we can come back to that in a minute. But maybe I’ll pause here and say something about Russell’s paradox because it’s actually fairly straightforward. Maybe not simple, but straightforward.
So here’s Russell’s paradox. You might think that given any property—like the property of being red or the property of being a basketball—there’s a set of things that has that property. The property of being red? Well, so there’s a set of red things. The property of being a basketball? Well, so there’s a set of all the basketballs. So that’s the basic idea: for any property, there’s a set of the things that have that property.
Now, think about some weirder properties. Think about the property of being a set. So there should be a set of all the things that have that poverty: that is, the set of all the sets. How about the property of not being a basketball? Okay. It’s a big set, so there’s the set of all things that aren’t basketballs. Some sets are weirder than others. Here’s a property that some sets have, and some sets don’t have. It’s the property of being a member of itself. Think about that for a moment. What kinds of sets have the property of being a member of themselves? Well, the set of all sets—that’s a member of itself, because it’s a set. The set of all non-basketballs. Okay; it’s not a basketball, so it’s a member of itself. The set of all tables. It’s not a table, so it’s not a member of itself. So think about the world of sets as divided into two. There’s the bunch of them that are members of themselves, there’s the bunch of them—really, the more normal ones—that are not members of themselves.
So, now we have this property: being a member of yourself. Now, let’s think about the set of things that have that property. Okay, so this is the set of all those sets that are members of themselves. The set of all sets is in there, the set of all non-basketballs is in there, etc. Okay. Now, just one more complication. Think about a slightly different property: this is the property of not being a member of yourself. So which sets have that? The set of all tables, the set of all red things, stuff like that. Those sets are not members of themselves.
Ok, so that’s the property of not being a member of itself. Think of, now—we’re getting to the end here—think of the set of things having that property. The set of all things that are not members of themselves. Got it? Now, ask yourself this question: is that set a member of itself? And it turns out that you’re not going to get a good answer here. Let’s think through why you’re not going to get a good answer here. Let’s call this set ‘R’, ‘R’ for Russell. R is the set of all those sets that are not members of themselves. And ask yourself this: is R a member of R? Well, if R is a member of R, then it must have that property that all its members have, namely not being members of themselves—oops! That means if R is a member of R, then R is not a member of R. That won’t work. Okay, so R is not a member of R. Good. But you can’t take a deep breath yet, because if R is not a member of R, then it has that property that all and only the members of R have. So, it is a member of itself. From the supposition that it is a member of itself, we reached the conclusion it isn’t a member of itself. And from the supposition that it isn’t a member of itself, we reached the conclusion it is a member of itself. So there is no way this set can exist. It can’t be a member of itself and it can’t fail to be a member of itself.
The reason this is called Russell’s paradox is that that basic principle—that for every property, there is a set of things having that property—that principle is what we just used to reach what philosophers call a contradiction. The contradiction is a claim of the form: some claim and its negation. So the contradiction we reach here is that R is a member of R and R is not a member of R. That’s the contradiction. The fundamental principle led to a contradiction, so the fundamental principle is false. And that principle was: for every property, there’s a set of things having that property. Frege thought that fundamental principle was true, and that was really, really important to the whole way he analyzed arithmetic. And so, now that that fundamental principle turns out to be false, Frege thinks there’s no logical principle from which you can prove the existence of anything like sets. And you needed to do that to do logicism the way Frege wanted to.
Jaime Edwards:
So Frege took this paradox to be devastating to his own
account of logicism, but Russell did not. Russell was working on a logicist
program as well, so why didn’t he find this to be devastating to logicism in
general?
Patricia Blanchette:
Good. For Frege, the idea was, again, to reduce
arithmetic to the kinds of obvious principles of logic that we were talking
about at the beginning of this podcast. Things like ‘if I have a black hat and
red shirt, then I have a red shirt’. Things that were so obvious that you
couldn’t question how we know them. We just know them just through very simple
reasoning capacities. Frege wanted the principles at the very basis of his
system to be that obvious, and in order to call them ‘principles of logic’, they
had to be that obvious. That was his idea.
When Russell’s paradox comes along, Frege realizes (and Russell realizes) that the principles Frege thought would work won’t work. Frege thought: and there aren’t any other principles that are sufficiently obvious that will work for generating arithmetic. Russell thought: well, it’s still a really interesting question to see what kinds of basic, fundamental principles we can use to generate arithmetic.
So what Russell does is: he replaces the problematic principle that Frege had with a few more principles, which turn out to be not problematic. That is, they don’t lead to paradox. And from them, you can do something like what Frege was trying to do. That is to say, you can deduce a bunch of—not all of, interesting—but you can deduce a bunch of mathematics from the fundamental principles that Russell took as the basis of his system. The big difference here is that those fundamental principles that Russell takes as the basis are principles that don’t have any obvious justification. They’re not obvious in and of themselves. Russell took them to be very, very interesting principles, and they are very interesting principles for lots of reasons, but what justifies them basically is the fact that they’re enough to give you mathematics. They’re not as obvious and self-evident as Frege wanted his principles of logic to be.
Matt Teichman:
So Russell unearthed this difficulty in Frege’s original
version of the logicist program. And it turned out there were some problems with
Russell’s solution to his own difficulty that he raised with logicism, as
well. Where does that leave us today? Is the logicist program mostly of
historical interest? Does it have something to teach us about the nature of
mathematics, the nature of logic, the nature of human reasoning, or
justification?
Patricia Blanchette:
Well, let me start by saying a couple things about
why people don’t follow Russell’s logicism. Why Russell’s project, that he
pursued with Alfred North
Whitehead in a three-volume work
called Principia
Mathematica that
was published from 1910 to 1913—why people don’t think that that established
the truth of logicism.
And here, there are two big issues. One issue is that, as we were just talking about, the fundamental principles (or, as we called them, the axioms) that Russell and Whitehead use at the basis of their system don’t strike most people as principles of logic. They are, again, useful and interesting axioms, but they’re not purely logical. Then, a much more probably historically earth-shattering thing happened—which is that Kurt Gödel proved what’s now called the incompleteness of arithmetic. And the incompleteness of arithmetic shows us, in a nutshell, that there isn’t any way to take a manageable collection of axioms—of the kind that Frege wanted, of the kind that Russell wanted—and use them in such a way that you would be able to prove from them every truth of arithmetic. It turns out—interestingly, and really, very surprisingly—that the truths of arithmetic are really an incredibly unmanageable collection of truths. Which is to say: there is no set of axioms from which all of the truths of arithmetic are provable. Which is a really cool, exciting, and really, I think, rather stunning thing to discover about arithmetic.
Now, what does that have to say about logicism? Logicism, the thesis that the truths of arithmetic are truths of logic, doesn’t actually require that the truths of arithmetic are provable from a manageable collection of axioms. After all, it just requires that the truths of arithmetic are truths of logic—and maybe the truths of logic aren’t provable from a manageable collection of axioms. Nevertheless, the way that Frege was trying to prove logicism, and the way that Russell and Whitehead were trying to prove logicism, was: to take a manageable collection of logical axioms, and prove from them a manageable collection of truths from which they thought all the truths of arithmetic could be proven.
And that’s the step that turns out to be wrong. There isn’t a manageable collection of arithmetical truths from which the rest of the truths of arithmetic can, in principle, be proven. So what the incompleteness theorem does is: it shows us not that logicism is false, necessarily, but that the straightforward way of trying to demonstrate the truth of logicism will not work. So that is something that, in fact, Frege never knew about. He died in 1925, and this result comes about in 1931. So we don’t know what Frege would have said about it, but he would have found it, I think, somewhat surprising as well. And indeed, I think Russell and Whitehead did as well.
So that’s one reason that you can’t pursue logicism today in quite the way that either of our players here—Frege, or Russell and Whitehead—were trying to pursue it. It also means that if you think of the truths of logic as what you can prove from some manageable collection of axioms, then, in fact, logicism is false So, in light of the incompleteness theorem, you can only be a logicist if you think that the truths of logic are an interestingly unmanageable collection of truths.
Now, there’s the other question you asked, which is: what’s the status of logicism today? There are some people who think, for those kinds of reasons, that logicism really ought to be abandoned—that it’s no good. And that may be because they think of the truths of logic as pretty manageable, and so, the truths of arithmetic are just not going to be reducible to the truths of logic.
On the other hand, there is a research project ongoing now which has earned the name ‘neo-logicism’, which is a research project inspired by a book written by Crispin Wright, called Frege’s Conception of Numbers as Objects. And that project attempts to resuscitate Frege’s research program, by trying to demonstrate that the truths of arithmetic can be deduced—not, maybe, from principles of logic—but from principles of logic plus a principle that just kind of explains what numbers are. And indeed, it does so in a way that Frege would really have liked.
The principle that they use is the one that we actually talked about a few minutes ago. It’s the principle that says this: the number of—now, I’m going to use the letters ‘f’ and ‘g’, so that ‘f’ might stand for ‘people at the table’ and ‘g’ might stand for ‘chairs at the table’, or ‘f’ might stand for some other property, and ‘g’ for some other property. We want to know: under what conditions are there just as many f-s as there are g-s? Well, as we talked about last time, the basic idea is: there are just as many f-s as there are g-s if you can match up the f-s one-to-one with the g-s, without any overlapping, without leaving anything out. Remember our example of the people, and the chairs, and the people sitting in the chairs.
Okay, so here’s the basic principle that neo-logicists use. They say you can explain what numbers are by saying this is the fundamental truth about numbers: the number of f-s equals the number of g-s if (and only if) there’s a way of matching up the f-s and g-s one-to-one with each other. And it turns out: that principle is remarkably rich. If you use that principle, plus other principles of logic, you can get a lot of arithmetic. Indeed, you can get what we call the Peano axioms. People think that’s enough to show something like logicism is true. Because that principle that I just gave you, even if you don’t think of it as a truth of logic—it’s not as obvious as the principle of ‘if I have a red shirt and a black hat, then I have a red shirt’—nevertheless, it’s something you can know without going out and measuring anything or counting anything in the world. It looks like something you can know non-empirically. It looks like it’s kind of just explaining what we mean by ‘number of’, or something like that.
So the neo-logicist project is the project of trying to make the case—by giving lots of arguments and some interesting proofs—make the case that you can, in fact, ground arithmetic in that principle plus logic. And the neo-logicists think that this tells us something very interesting about the nature of mathematical knowledge—that it is something like what Frege thought it was. So that’s an ongoing research project. Also, there’s a lot of historical interest in the kinds of reductions that Frege managed to give, just for mathematical reasons—not from the point of view of reducing mathematics to anything else. So that’s another sort of historical legacy here.
Matt Teichman:
So were mathematics and logic the only topics that
Gottlob Frege wrote and thought about, or did he work in other areas of
philosophy as well? How did this fit into his general philosophical picture?
Patricia Blanchette:
Oh, another nice question. In fact, a lot of
philosophy students know about Frege and they don’t even know that he did
philosophy of mathematics or logic, because a lot of his work has turned out to
be very influential in a field called the philosophy of
language.
Philosophers of language are interested in questions about how it is that you and I use language to communicate. How is it that I get ideas from my head into your head just by making these kinds of sounds or writing marks on paper? And they’re also interested in how our language gets to be ‘about’ things in the world. Notice for example that you and I can talk now about the Eiffel Tower, even though it’s not here. Somehow, the words that I used got you to think about this object—and I managed to say something about this object—that’s very far away from us.
So, these are questions that many philosophers have thought of as very interesting, and they all come under the title of ‘the philosophy of language’. Frege was interested not so much in the language that we use to talk about things like the Eiffel tower—though he did talk about that kind of language—but he was extremely interested in the languages that we use to do mathematics. So, for the kinds of reasons that we’ve been talking about, he was interested in analyzing the meanings of various kinds of mathematical statements. And then, he found that this was difficult to do without coming to a better understanding of what meaning really is. You might ask yourself this: when you use a word, what exactly does the word mean? Like the word ‘the Eiffel Tower’–okay, it’s three words, but anyway, the phrase ‘the Eiffel Tower’—does that mean the big tall thing over there in Paris? Or does it mean the idea that you and I have when we use the word? Or, like, what, exactly?
Frege had several things to say about that. The fundamental thing that Frege tends to be remembered for when it comes to language is that he thought that every different piece of language has two different kinds of meaning. He called the first one the sense and the second one the reference. If you take a phrase like ‘the Eiffel Tower’, the reference of that phrase is that building; that tower there in Paris. It’s an actual object. Whereas the sense is a rather more difficult thing to describe thing. It’s what you and I grasp when we understand the word. So if someone is speaking to you in a language you don’t understand, what you’re not doing, thinks Frege, is associating the right sense with the words. So meaning—in the sense of what you grasp when you understand the language—that’s the sense of your words. Meaning in the sense of the stuff out there that you talk about is the reference of your words.
And this holds even when your words aren’t about physical objects, like the Eiffel tower. Let’s think of two phrases. Here’s one: ‘two plus two’. And here’s another one: ‘four’. Those two phrases, interestingly, have what Frege thinks of as the same reference: they both refer to the number four. But they do something that’s quite different. Because when I say, ‘two plus two equals four’, I say something quite interesting—well, at least a little bit interesting. More interesting than when I say, ‘four equals four’. So the phrase ‘two plus two’, in some sense, has a different meaning than the word ‘four’.
Here’s Frege’s idea: the phrase ‘two plus two’ and the phrase ‘four’ have different senses, though they have the same reference. He thought that it was very important to explain how sense is related to reference, how sense is related to understanding, how reference is related to understanding, and things like this. And cashing out all those ideas took quite a bit of work on his part. He wrote a number of essays that have become extremely influential in the philosophy of language. And a lot of work done today in philosophy of language really can position itself—that is to say: people can explain where they stand on various issues by saying things like, ‘Well, I agree with Frege about this, but disagree with Frege about that’. His ways of thinking about language have had a lasting influence, even today, on the ways that we think about issues in the philosophy of language.
Matt Teichman:
Patricia Blanchette, thanks very much for joining us.
Patricia Blanchette:
You’re welcome. Thanks Matt, thanks Jaime. This
was fun.
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