 Hello, I'm JJ Joaquin and welcome to Philosophy and What Matter is where we discuss things that matter from a philosophical point of view. Now today, we'll talk about Gottlob Frege. And Gottlob Frege is recognized as one of the founders of analytic philosophy. Now his contributions range from logic to the philosophy of mathematics and the philosophy of language. And guiding us through the philosophy of Frege is my dear friend, Patricia Blanchett, McMahon, Hank, Professor of Philosophy at the University of Notre Dame, and the author of the book, Frege's Conception of Logic. So hello, Professor Blanchett, welcome to Philosophy and What Matters. Hello, JJ. Okay, so before we get into our main topic, let's first discuss your philosophical background. How do you get into philosophy? You know, I had the good fortune to go to one of those universities that allows you indeed requires you to take courses in lots of different fields. I stumbled into philosophy. I actually took a logic course because I thought it sounded interesting, and it was, and it was interesting enough that I took the next few things taught by that professor and suddenly after some time I had more philosophy courses than I had of the things I had intended to study. So I just kept doing it. And yeah, like so many people, I fell into it and I loved it. And that was the end of the story really. So who was this professor? Yeah, so my first philosophy professor was a junior professor by the name of Paolo Dow. He didn't stay in philosophy, but he was my introduction to the field. Okay, so you did a lot of logic during that time. So what sort of logic were you introduced to? My first logic course was actually an introduction to set theory. And I thought that was very interesting. And so then I took just a couple of courses in modern first order logic up through completeness and Livingham-Skollum theorems as an undergraduate. That was that was all I did really in logic. There wasn't very much to do. I did a little bit of computability, but not much. So who influenced you to pursue a career in academic philosophy? You know, I think probably every philosophy professor I had as an undergraduate was really important to me. I thought that the idea of thinking hard about texts and about philosophical ideas looked fascinating and exciting. And then it looked fascinating enough to get into graduate school. And then I had some, some, some very formative mentors there, John Etchamendi, John Perry, John Bar-Wise. These are all people whose work meant a lot to me and whose mentorship meant a lot to me. And yeah, I think they sort of solidified my thought that philosophy was a really, really interesting way of life. Okay, so let's get into our main topic. What did you get into the philosophy of Frege? Why specialize in Frege? That's a nice question. I never intended just to work on Frege so much, but I was interested in the relationship between logic and language. And in particular, I was interested in logicism. So I was interested in this theory that mathematics is in some sense reducible to logic. And so I thought, well, so I should, I should read some logicism. So I, so I read Frege and then I became really interested in sort of pursuing this idea to try to see where logicism went. So there was a point at which I thought I would write a dissertation on logicism in general and do a little bit of Frege, a little bit of Karnat, a little bit of Russell and Whitehead. But the Frege sort of took over and it became all Frege. Also, a secondary interest was that I was interested in the philosophy of language and I thought if I wanted to know about Frege's views about language, I should really understand his views about mathematics because they didn't seem to be distinguishable in my view. So yeah, that's why Frege. Before we get into his mathematics, his views about logicism and his views about language, let's start with logic first. So what were his contributions in the development of logic? So Frege was one of the three people who gave us modern logic in the sense of a quantified system of logic. The other two people were Giuseppe Peano and C.S. Perce. Frege's system was the, in some ways, the most well developed, the most thoroughly developed system of logic. So what we get with Frege is what you now think of as first order and second order logic. That is a logic that involves the first order and the second order quantifiers and their interaction with the ordinary connectives of negation and conjunction and material conditional. Frege is one of the first people who puts together a system that is to say a formal language that has rules of formation for its sentences and rules of inference and axioms to give logical proofs to prove one sentence on the basis of other sentences. Prior to Frege's work, the proofs were not nearly as well. There was no way to formalize a proof. That is, to turn rules of proof into rules that were syntactically given. And the interest of that is, well, it's really interesting. But the interest of having formal systems of logic has changed over time. For Frege, the reason to have a formal system that is a syntactically defined bunch of rules was that this made it possible to systematically check whether the proofs were good proofs. So instead of relying on intuitive ideas of what follows from what, you have a very, very rigorous system within which you give proofs that follow very, very clear rules, and then it's clear that you can check them. But once Frege gave us this idea of syntactically specifying rules of inference and axioms for system of logic, once we have that idea, then this turns out to be a very fruitful thing for a couple of reasons. It turns out to be very fruitful for what became meta mathematics and turned into the work that Hilbert and Goethe gave us. And it also became extremely fruitful for what became computer science, because what you have with computer languages, computer science in general, is a system by means of which everything can be specified syntactically all of your rules of transformation. So what you get with Frege is the beginning of modern logic and then beginning of what eventually becomes computability theory and computer science. Okay, so that's a long history from Aristotle. So someone has said that logic really started in 1879 with the publication of this book, The Griff Schrift. So why is this a monumental work in logic and in philosophy in general? So I should say one, one probably should not say that that's where logic starts logic. Certainly Aristotle was doing logic in some sense. But modern logic arguably starts here and I mean one shouldn't forget person piano as well who were also extremely influential and doing very, very similar things. So what happens in the Griff Schrift is you get, yeah, the two things that I mentioned, namely a systematic use of quantifiers, which we just didn't have before. So this is this is brand new and it's huge, and the, the syntactic specification of the rules of inference and the axioms in the system. The thing about the quantifiers is that prior to having a formal system of quantified logic, you can't give a nice treatment of inferences like this one. Every even number is greater than at least one odd number. Mathematics is an even number. Therefore, there is at least one odd number such that six is greater than it. That's a fairly rudimentary piece of reasoning that you've got to be able to make sense of in order to do anything sophisticated in order to do mathematics in order to reason about anything really. And unless you have a really nice way of using quantifiers, you can't actually talk about what makes that inference valid and what makes other inferences that are superficially like it invalid. So anyway, with the Griff Schrift you get the first nice systematic treatment of quantifiers. So this is a real development from like Aristotle even can perhaps. So, so prior to 1879. You had some systematic treatments of various forms of inference, so various syllogisms that Aristotle dealt with, and various more sophisticated forms of inference dealt with by various medieval scholars. But what you didn't have was enough sophistication in the systematic inferences dealt with in logic to do mathematics. So the, the kinds of reasoning that mathematicians use every day to think about numbers were just too rich to be dealt with by the kinds of inferences that were nicely systematized by Aristotle and by later peoples. So there was just a big gap between what counted as good reasoning in a mathematical context, and what counted as the kind of reasoning that we had any systematic treatment of prior to 1879. So, is it safe to say that Frege's system is really developed because he wants to give a foundation to mathematical reasoning in this making. So that's the main target. So, so the grift shift the first system that Frege introduces opens with the following question. How much of mathematics is founded on pure logic. And for him, that's kind of the driving question and it's a question that he doesn't think that he answers in 1879, but, but he thinks that in order to begin to answer the question, we need to get much more clear about what we mean by pure logic. And we need to have a system of proof that's really, really rigorous. And the reason for that is that in order to answer the question, how much of mathematics is grounded in logic, you need to be able to start with some mathematical truth and then show and prove them from purely logical truths. So you need an account of what you need a list of some purely logical truths, and a list of some purely logical rules of proof, and see if you can prove the fundamental truths of mathematics from those. So that's what he's beginning to do in the grift shift. But why was he considered as one of the founders of analytic philosophy. Ah, well. So, I think really because of maybe three things. The first is what we've already discussed he gave us a system of logic. And so logic has turned out to be a really core part of analytic philosophy. It's useful for. Yeah, lots of reasons right so a core part of analytic philosophy is a view about what it is for certain things to follow from other things of what it is for truth to be analytic. Which parts of mathematics are grounded in logic and all of those turn on views about logic. So, so having a system of logic is already critical to, to modern analytic philosophy. Another aspect of Frigga's work that we haven't discussed yet that's essential to analytic philosophy is Frigga's philosophy of language. So Frigga, as, as anyone who's who's done a first course in the philosophy of language knows Frigga is one of the founders of a particular way of thinking of language and its relationship to our thought. So, so Frigga thinks we can ask questions and give systematic answers to those questions regarding the meanings of the sentences that we use and I can say more about that in detail later but, but the second way in which Frigga is kind of a founder of analytic philosophy is that he's a founder of contemporary philosophy of language. Everyone who works in philosophy of language now has to either agree with Frigga or disagree with Frigga about certain fundamental views about language. And then, thirdly, Frigga has a view about, about conceptual analysis that is really important to how we do analytic philosophy today. He thinks that you can buy careful analysis of concepts as these occur in, in ordinary science in ordinary discourse and in mathematics in particular by careful analysis of our concepts, we can learn what kinds of principles ground the kind of discourse in question. So, actually, let me mention a fourth thing. One thing that's really, really important is a certain anti-psychologism that Frigga has. That is to say a lot of Frigga's views about language and about mathematics involve the following view that the truths of mathematics and the truths about, well let's just say the truths of mathematics don't in fact have much to do with the way people actually reason. His anti-psychologism is the following view that psychology and mathematics are radically separate. And this theme runs through a lot of his views about, about knowledge and about mathematics, about foundations of science. And so that's another theme that's been very, very important to contemporary analytic philosophy. Okay, so we touched on his views on the philosophy of mathematics, specifically his notion that mathematics is reducible to logic, or at least the foundations of mathematics would be a kind of logical foundation. So this is the logistic view. So you mentioned about the psychologist view, the psychologist, which Frigga and Russell, other guy who's the founder of logistic, were against. But what is their view all about this logistic view and what were its competitors. Okay. So I should say, first of all, let me just clarify one point. Frigga doesn't think that all of mathematics is reducible to logic. He excludes geometry from this. So Frigga thinks that Kant is right about geometry, that geometry has some fundamental truths that are known to us via pure intuition. And so, so geometry is, is not part of the logistic project for Frigga. So you asked, what are the competitors? Frigga took himself to be arguing against, I think really three different views. One view was Kant's view. So, so Kant's view about arithmetic is similar to his view about geometry, that is to say that the fundamental truths of arithmetic are much richer than something that could be founded in pure logic that one needs in addition to logical reasoning, one needs pure intuition. So intuition of space for the case of geometry and presumably for time of time for the case of arithmetic. So Frigga wants to reject that view. He thinks that pure intuition plays no role in the foundations of arithmetic. By arithmetic here, by the way, he means not just what we might call arithmetic, which is sometimes taken to be just number theory, but also real analysis as well. So the complete theory of real numbers. Okay, one of the views then that Frigga took himself to reject was Kant's view. Another view that he took himself to reject is a pure empiricist view of the kind that John Stuart Mill arguably held. I mean, I guess Mill clearly held an empiricist view what exactly the view was I think it's hard to determine, but Mill thought that people come to know the fundamental truths of arithmetic via the sensation. So for example, you put three pebbles in a pile and five pebbles in a pile and then you count them together and you have eight pebbles. So the meaning of your understanding that three plus five equals eight and Frigga took that view to be clearly false. He thought it wouldn't work for large numbers that certainly wouldn't work for infinite numbers and he thought it was a very bad way to think about our knowledge of arithmetic. And then the third view is, I think sort of a mush of different views. It's hard to attribute any one person but the mush is just called psychologism. That is to say, any view that says that part of the content of arithmetic is a bunch of claims about how people think that we'll call that psychologism and Frigga thinks that's false. No truth of arithmetic depends in any way, Frigga thinks, on the ways that human beings actually reason or actually prove things to themselves. So he thinks he thinks that any view that says that part of the content of our claims about infinity have to do with the ways that people think about infinite collections has to be false. And fundamental he has his idea is that the truths of arithmetic would have been true, no matter what kinds of brains people had, no matter whether there were any people anything like that they are as independent of us as claims about rocks and trees are independent of us. So, so those I think are the main views he took himself to reject. So we have intuitionism, the Kantian version where the truths of mathematics are products of the categories of the understanding perhaps of time. And you have empiricism that you need to look at the world so to speak and look at whether two rabbits and another two sets of rabbits would give you four rabbits. Well, that's not the case. Right. And finally, you have psychoticism that the truths of mathematics would depend on how people actually think about them. Yeah. So what is his theory, what is this logistic and it's, it looks like that logistic and implies the kind of realism about the truth of mathematics and the truths of mathematics are out there, so to speak. But how do we know those things. We know those things via our capacity to reason logically. That is to say, why are capacity simply to to reason in a way that avoids contradiction and to reason from premises to to the things that follow from those premises. So the first thing to think about with respect to logic, I think to understand what logic is for someone like Frege is to think about the fact that all of our reasoning involves following out the consequences of things that we believe. So, so if you believe that every politician is corrupt, and you believe that Smith is a politician. Well, you really ought to believe that Smith is corrupt. If you fail to draw the conclusion that Smith is corrupt, you have failed in some way. And indeed, if you, if you affirm, in addition that Smith is not corrupt, then you've engaged in contradiction. So the fundamental human capacity to to reason in accordance with logical entailment that is to say to to draw a conclusion from premises when that conclusion really in fact follows from those premises. That's a human capacity that obviously we all have thinks someone like Frege why I presume thanks everybody but anyway it's a capacity that we all have because we can all reason. We can critique each other's reasoning so it's not just that we happen to say one thing after saying another. It's that sometimes we do so in a way that's justified, because the thing we've inferred follows. And sometimes we do so in a way that's not justified because the thing we've inferred doesn't follow. So that relationship the relationship of following from is something that Frege thinks we know something about. How is it that we know something about that. We are just able to see consequences when they are simple enough. So once you grant that, then it also looks like you have to grant that that people are able to use that capacity to get capacity to reason logically. We're also able to use that capacity to know that some things are true, not just that if something is true than something else is true but also to know that some things are true, like some conditional claims. We can make that inference that I just gave you into an if then claim we can say if all politicians are corrupt and Smith is a politician then Smith is corrupt, and there you go there's a truth of logic. Frege isn't particularly interested in the psychological question of how people do stuff. But he does think that it's clear that there is a logical source of knowledge, and that is to say it's that capacity that we have to reason in accordance with logical principles to draw inferences and to recognize contradictions, things like that. And he thinks that's that's something that gives rise to knowledge of truths, and it's clearly something that it has to be independent of our capacity to know things via sensation, and independent of our capacity to know things by intuition that's different from those capacities. So that's the beginning, and Frege thinks that if he can show you that the fundamental truths of arithmetic are provable using just the kinds of truths that you can come to know via your logical capacity. That's going to be done. He will have shown you that you don't need anything like pure intuition, and you don't need anything like sensation, in order to come to know the truth of arithmetic. That's the story. And then the details are where it gets hard. Yeah, that's the next question was he successful in reducing mathematics arithmetic to logic. Sadly, no. I think he would be successful so there's this book that you have in front of us the good login that the foundations of arithmetic was a really fabulous work and in it, Frege introduces his audience to the kind of conceptual analysis of arithmetic that he thinks will be important to following out his logistic project and he also begins to introduce his audience to the kinds of proofs that he will give. His idea is that he will show that logic isn't as true by doing two things. First thing, give a really thorough analysis of what arithmetic statements mean. So we can see clearly what we're dealing with. Secondly, show that once the statements are understood in this way, we can prove them from purely logical premises. So the foundations of arithmetic book written in 1884 is an attempt to do the first part of that, and to sketch the second part of that. So he does that and it looks very promising it looks very good he gives us a very rich, it turns out very influential way of analyzing our mathematical truths, and he begins to give the proofs, and it looks like that will work out to Then in 1893 he starts to do the really, really hard work of giving the proofs in excruciating detail. He has a nice new formal system which is richer than the system of the griff shift that we spoke about earlier. And he translates his analysis of arithmetical truths into the language of this new system, system of Grungis etsa. And he tries to give, indeed he gives unbelievably rigorous step by step by step by step by step proofs so that after many, many, many pages you get things like one is the successor of zero takes a long time. And that's the piano action. That's the first one. Well, it's the second one. He gives us. Yeah, he gives us a version of the band of axioms, and it looks like he can prove them from purely logical principles it looks, it looks beautiful. But it turns out, doesn't work so well. So in 1902, for a good gets a letter from Bertrand Russell, explaining to him that you can prove a contradiction in prego's formal system, and this is, this is really bad this is as bad as anything can be for a formal system. So let me remind you, the formal system was a, a bunch of fundamental logical truths and fundamental logical rules of inference for getting new truths from old truths. So using the axioms, the fundamental truths and the rules, using those you ought to be able to prove only truths of logic, everything you prove should be a truth of logic. And that's why proving the truths of arithmetic in this way would show that the truth of arithmetic truths of logic. But alas, as Russell showed you can prove a contradiction so that means that you can prove something false. That means something has gone badly, badly, badly wrong, either one of the axioms, or one of the rules has to be mistaken. And it was very clear which axiom it was there's only one candidate and it was, it was called basic law five it's a particular thing that Frege thought was a truth of logic, and isn't, we now know it's not a truth of logic. So Frege's entire system required this thing, basic law five to be a principle of logic and it isn't. So the whole thing falls apart. And what's really painful is that both parts of what I've just described as the two part project fall apart part one was the analysis of arithmetical truths and part two was the proofs of the thus analyzed truths. And because basic law five, the thing that leads to the contradiction is the really critical thing. It's the critical principle that governs things called value ranges, which are a lot like sets is the critical principle governing those, because that's the principle that's false. Well, Frege's analysis of arithmetical claims turns out to be very, very problematic because he analyzed arithmetical claims in such a way that they all have to do with value ranges. These things that are success. So, so the analysis turns on value ranges, and value ranges don't behave the way Frege said they did. So the proofs are bad. So the whole thing falls apart. So, yeah, so Frege did not prove that logic system is true. And, and it turns out for other reasons. There isn't a good way to fix Frege's system. It may be that logic system is true. But it certainly cannot be proven to be true in anything like Frege's way. So is this like the fruit of the poison tree principle going on. Well, Yeah, I don't know. So, so it was very bad. Certainly. Yeah, it was, it was terrible for Frege. His whole thing crashed and burned. On the other hand, out of this crashing and burning, we've had a massive flowering of things. So, so what we learned was that it wasn't just Frege's particular idiosyncratic way of thinking of value ranges that was problematic. It was the way that everybody had thought about collections, sad like things or ranges of values of functions. So for much of the 19th century, than reasoning carefully about collections of numbers, about collections of functions, about collections of functions, of functions, et cetera, in ways that, in the background, used something like Frig's basic law five, some principle, like what we call now the principle of extensionality for collections of things and for, yeah, as I say, ranges of values of functions. So what we saw with Russell's paradox and with some paradoxes that came up just a couple of years before that, though Frigga didn't know it, that were discovered by Cermelo in about 1900. What we see here is that a way of reasoning that was really important to mathematics at the time turns out to have been unreliable. And so it became clear right around 1900, 1901, 1902, that something different had to be done that if we were going to keep reasoning, both in the way that Frigga had been reasoning and in the way that everybody who'd been talking about sets, especially Kantor had been reasoning, turned out we needed to fix the system. We needed to figure out how to safely reason about collections of things and courses of values of functions. And that's where axiomatic set theory comes in. So axiomatic set theory is a way of laying down principles which will give us a good fruitful safe theory of collections, which allows us to do the mathematics that we want to do and avoid the paradoxes that befell Frigga's system. Unfortunately axiomatic set theory is it doesn't meet the conditions for being pure logic than Frigga thought. And so even though now we know how to do a lot of what Frigga wanted to do by using an axiomatic system of set theory, there are a couple of reasons to think that this isn't a vindication of Frigga's logistic project. One reason, the main reason is that it looks like the principles of axiomatic set theory do not count as, they're just not purely logical. Yeah, exactly. Okay, so yeah. Yeah, so let's not get into the Russell paradox because yeah, Russell's paradox has been discussed in many other books and podcasts. Let's just ask the question, is the logistic program still alive and well and contemporary philosophy of mathematics? There is interest in logisticism in contemporary philosophy of mathematics. The most active project is what's called the Neo logistic project, which follows on from some work by Crispin Wright and Bob Hale. So there's a book by Crispin Wright called Frigga's Conception of Numbers as Objects in which Wright suggests that we can resurrect a logistic program that is in some ways like Frigga's. So let me just say something about why it's not the same as Frigga's and why it's interesting nevertheless. So Frigga's strategy was of course to start with purely logical truths and to prove all of the truths of arithmetic. The Neo logistic strategy is to start with something different. It's to start with some logical truths and then a thing called Hume's principle, which is a principle that just essentially says this, the number of, well, if we have some two properties like the property of coffee cups on my desk and the property of pens on my desk, we can say the number of coffee cups on my desk is identical with the number of pens on my desk. So there's a statement of identity and we wanna know under what conditions is that true? And here is Hume's principle and instance of it. The number of coffee cups on my desk is identical with the number of pens on my desk if and only if there is a one to one matching up of the coffee cups on my desk with the pens on my desk. So that seems good. And in general, the idea is the number of, we'll call our two properties F and G, the number of F's is identical with the number of G's if and only if there's a one to one matching of the F things onto the G things. This works for finite and for infinite collections of things. So that's very nice. This principle called Hume's principle played a large role in Frig's own project. He thought that he could prove this principle from purely logical principles. You can't. And so the neologesis strategy is merely to start with that principle as one of the foundations of the project. Start with that principle plus logic and then prove the piano axis. Okay, so this differs from Frig's project in one fundamental way, which is that Hume's principle doesn't look like a truth of logic. And indeed, I think it's clear for various reasons that would not have been thought of by Frig as a truth of logic unless you could prove it to be one in the way that he tried to. Once Frig sees that his axiom, basic law five, is not a truth of logic, it would follow from that, I think, that Hume's principle and his view is not a truth of logic because Hume's principle is very, very, very like basic law five. And in a case, so there is debate about this these days whether or not one should think of Hume's principle as something like a truth of logic. So first clear difference between the neologesis project and Frig's project is that Frig's project wanted to start with fundamental truths of logic. The neologesis project doesn't care so much about whether the fundamental principles are principles of logic. They're interested in starting with Hume's principle because it's special in some way. It's something like analytic, so that's good. Another big problem though is that when Frig was writing it seemed that it would be possible to prove the paeno axioms from truths of logic and thereby guarantee that every truth of arithmetic was a truth of logic because it seemed that one could prove every truth of arithmetic from the paeno axioms. So sort of three steps, start with logic, prove the paeno axioms and then use those to prove any truth of arithmetic you want. But we know because of Goedl's incompleteness theorem the first one, that this won't work. It's not true that from the paeno axioms you can prove every truth of arithmetic. Indeed, it's not true that from any manageable collection of axioms you can prove every truth of arithmetic. So another thing that distinguishes the neologesis project from the Frigian project is that it is not an attempt to prove a core collection of mathematical truths from which one can in turn prove all of the truths of arithmetic. There is no such core collection of truth. The neologesis project is instead to prove a core collection of arithmetical truths that suffices in a very different way for all of the truths of arithmetic. And for those who've done some logic it suffices in the sense that every model of those axioms, of the second order paeno axioms is a model of every truth of arithmetic. And I think that there isn't any way that Frigian would think of that relationship between the paeno axioms and the rest of arithmetic as supporting the logistic plan. So that's another very important difference between them. For Frigian, the important connection between the paeno axioms and the truths of arithmetic was going to have been proof theoretic. You can prove the one from the other. Whereas for the neologesis project it's a very different relationship. It's a model theoretic relationship. It's a model theoretic relationship, yep. Yeah, but there's another movement going on around known as paraconsistent mathematics. I think what they're trying to do is, okay, let's abandon all hope about consistency and let's just accept that there are contradictions within your system and have a logic that guarantees that, well, your contradictions won't be due any other proposition. What do you think of that kind of project? I think it's really interesting and I think it's both technically and conceptually fruitful. I don't think it's easy to think of it as giving us what Frigian wanted, which is an account of those principles such that our knowledge of them is both sufficient for mathematics and counts as obviously correct principles of reasoning. Because you have to be very careful when you're doing paraconsistent logic to make sure that the inconsistencies that arise in some parts of your system don't ever touch the parts of the system that you really care about. And the kind of care that you need for that is, again, a really, really interesting project, but it's not a project that turns on the claim that the principles you use in being so careful are self-evident logical truths. So, yeah. So, also, I don't like contradictions myself. So I think it's conceptually really interesting, but for my own part, and this is one of the reasons that I don't know very much, to be honest, about the paraconsistent approaches, though I wish I did, it's that I have kind of a Frigian picture of the connection between truth and contradiction and the connection between truth and negation. And on that picture, it isn't possible for there to be a contradiction that's true. Indeed, on that picture, the claim that some sentence is true conflicts with the claim that that sentence is a contradiction. And so it's really hard from this point of view to understand what it would be for a contradiction to, in fact, be true. It's hard to get a grip from a Frigian point of view, indeed, from my point of view, too. Hard to get a grip on what a contradiction is actually saying. So in any case, yeah, that's one reason that the paraconsistent approach, I think, is pretty radically different from anything that Frigian would think of as a logical approach. Okay, so most students are introduced to Frigian's philosophy, through his works on the philosophy of language. So could you give us an overview of his main ideas here and how these ideas connect with his overall philosophy? Yeah, yeah, good, nice question. So you have the classic essay up here, On Sense and Reference, as it's called, which is in 1892. So here's a really interesting thing about the philosophy of language that I think one has to come to grips with in order to understand its role in Frigian's development. It's that the really, really astonishingly hard mathematical work that Frigian was doing to try to prove every truth of arithmetic in the world's most rigorous system is, it's a bunch of work whose first volume was published in 1893. These philosophy of language essays were published in 1891 and 1892. So it's not as if these were compartmentalized parts of Frigian's life. He was working very, very hard on his logistic project when he stopped to write essays about language. And so I think a really interesting question is, what is the connection here? And here I think is at least part of the connection. One aspect of Frigian's view of mathematics that we haven't talked about very much is that he thinks that arithmetical claims like two plus two equals four are true because these objects two and four have various properties and stand in various relations to one another. So he's not one of these people who says, you need to explain away this apparent reference to numbers. He thinks, no, we really do refer to numbers. When we say there are infinitely many prime numbers, we're making a claim that's just like saying, there are lots of coffee cups. We're just saying some things exist and the claim is true because the things exist. Okay, so given that background, think about the following two sentences. Three squared equals nine and nine equals nine. It looks like those sentences clearly say different things. And one way to become convinced of the fact that they say different things is that what would count as a proof of one does not count as a proof of the other. And Frigian's view is that what you're proving is not the sentences, but what the sentences say. So let me just explain that for just a moment. Frigian thinks that when you give a proof, you can give the proof in German, you can give the proof in English, you can give the proof in a fancy formal language, and it can still be the same proof. So mathematicians who speak different languages are still doing the same mathematics. Okay, so now when we ask, so what is it that three squared equals nine, what is it that that sentence expresses? And we ask ourselves, does it express the same thing as nine equals nine? When we notice that a proof of nine equals nine, which takes one line, won't count as a proof of three squared equals nine, we have to recognize that what's expressed by those sentences must be different. And so, so far so good. But then if you say, but wait, it's supposed to be the case that mathematics is about these particular objects. So three squared, that piece of language, just refers to the object nine. And the symbol nine just refers to the object nine. So it looks like the sentences cannot in fact, express different claims. They both just say, this object is this object. And so, so this I think is one way to think about the main motivation for Frig is probably, probably the thing for which he is most famous, namely the distinction between sense and reference. So Frig's view as it comes out in 1892, is that each sentence has two different kinds of meaning or each piece of language has two different kinds of meaning. And let's start with those little pieces of language, the symbol nine and the slightly more complicated symbol three squared. So he thinks that the symbol nine, it refers to, it has as its reference an object nine and three squared also has as its reference the object nine. So it's what we said before, those two little symbols refer to the same object, but they have different senses. They refer to the object in different ways. Frig is famous example of this is of course, not an arithmetical example because he didn't expect his audience to agree with him that numbers are objects. So his most famous example of this is the morning star and the evening star. Those two terms both referring to the same object, namely the planet Venus. And Frig illustrates the issue by pointing out that when you say the morning star is the morning star, you don't say anything very interesting, but when you say the morning star is the evening star, you say something that it would take astronomical research to demonstrate. And so in this case, again, we see that two pieces of language can refer to the same object, namely the planet Venus, but they can do so in different ways. The morning star, the phrase, the morning star and the phrase the evening star then have the same reference, but a different sense. So now when we look at a complete sentence, a sentence like three squared equals nine or nine equals nine, we can see that those two sentences, even though their parts have different references, the fact that their parts have different senses is important. So the sense expressed by three squared equals nine is different from the sense expressed by nine equals nine. Why? Because they have parts that have different senses. Similarly, the sense expressed by the morning star is the morning star, is different from the sense expressed by the morning star is the evening star. So there's his most famous innovation as it were, the distinction between sense and reference. The idea that words don't just stand for things, they do other stuff too. They have that. This is, I think, really important, in fact, to his view about the nature of mathematics, but he doesn't talk very much about this connection. It comes up here and there, but there's very little philosophy of language in his formal mathematical work and there are lots of interesting things people have to say about why that is. And there's not very much mathematics in his philosophy of language, though there's the occasional mathematical example. And I think the main reason for this is just that he didn't want in introducing the philosophical issues to complicate his argument against various forms of idealism and Kantian views and things like that. So there's the big thing, it's the distinction between sense and reference. There are other aspects of Frege's philosophy of language that have been extremely influential as well, and they are also connected with his philosophy of mathematics and logic. One of them is what's called the context principle, which has arguably been influential in all of analytic philosophy, and it's just this, it's the statement that, if you wanna know the meaning of a word, don't just look for a thing that that word stands for. As Frege said, don't look for the meaning of the word in isolation, but look at the contribution made by that word to the sentences in which it occurs. And one of the things that he says in the Foundations of Arithmetic is that, if you forget this principle, then you're likely to make the mistake of thinking that arithmetic is about ideas. And so let me just say why he thinks this, at least it might be why he thinks this, it's as follows. If you want to know what the number, let's just take the numeral three. If you want to know what that numeral stands for, that is to say, if you violate the context principle and you ask, what thing does this numeral stand for? And you look around, there are many things, it's hard to find anything. And so the only thing it could really stand for, if you're looking for a thing for it to stand for, is maybe an idea in your head. So then you get the picture that, okay, number words stand for ideas, and then arithmetic is all about our ideas, and then you're off to the races on a very bad theory, he says, Friggy. He thinks instead, if you want to know what the numeral three means, you should look at how it works in sentences. And you should look at how it is, what are the truth conditions of a sentence, like there are three pens on my desk. There's a use of the word three. And if you can understand how it works in that context, and you can also understand how it works in contexts, like three is the successor of two, if you can meld those together into a good account of the meanings of entire sentences, you will then know what the numeral three means. Okay, so the context principle again is the idea that, in order to understand the meaning of a word, you should look at the contribution that that word makes to the sentences in which it occurs, rather than looking for an object that you would have to find by taking the word in isolation. There's a lot of debate about whether Fraga maintains the context principle throughout his life. And my view on that is that he does, and that it's a very important principle, but there are reasons to doubt this. So there's a certain amount of back and forth. It's a, if one gets really into the weeds of Fraga's philosophy of language and scholarship on it, one has to have a view about this. But one thing that seems clear is that the context principle itself has had a lasting influence. I think it was very influential on Wittgenstein. It seems to have been very influential on Karnab. And so there's another piece of Fraga's philosophy of language which one needs to come to grips with in order to figure out what his kind of legacy was with respect to his views about language. I can't help but be reminded of Wittgenstein's quote, only in the nexus of a proposition has a name, has name has a meaning, something to do with it. And I think that that's as clear as nothing fancy, that's Fraga. Okay, so for you, what's the lasting legacy of Fraga's philosophy? It's complicated. So we talked about part of it already, that is to say the foundation that he gave us for modern logic, both making it clear how important quantifiers are and how you can deal with them systematically was huge. His view that in order to understand various kinds of knowledge, and in particular for him most importantly, knowledge of mathematics, you really have to understand how the language of mathematics works. That's an enormously important legacy. Not everybody agrees with Fraga's particular views about language, but I think it's fair to say that it would be hard to reject Fraga's thought that in order to understand what it is that we do when we come to know something, it's really important to understand what the words mean that we use to express that thing. So the integration between questions about language and questions about knowledge, I think is a lasting legacy of Fraga's, where he's not the first person who thought this, I think Plato thought this as well, but Fraga used this idea in a much more systematic way, I think than had been used before and was extremely influential in this. The idea that conceptual analysis can be done rigorously, can be defended, can be the source of insights, have a lasting influence in part because a lot of people want to reject it. So there's a sense in which the tradition that goes from Fraga to Karnat to Quine involves changes and rejections of Fraga's view about how fruitful conceptual analysis can be. So that's a way in which his legacy has been important. It has spawned arguments about the role of conceptual analysis. And I think the arguments are extremely fruitful and are alive and well. His views about mathematics and logic have been extraordinarily influential. So his rejection of psychologists was robust and scathing and made a lasting impression on mathematics and philosophy of mathematics. His views about axioms, which we haven't talked very much about, were also very important and influential. So there was a famous debate is not quite the right word, but as something like a debate between Fraga and Hilbert about the nature of axioms and the nature of theories and focused on the nature of consistency and independence proofs. And there's a sense in which Fraga was representing an older view and Hilbert was representing a newer and now dominant view. So it's easy to see Fraga as having views here which have simply been eclipsed. But I think that that's not quite right. I think that Fraga, that the Fraga-Hilbert debate gave us an opportunity to do, well, I think they themselves articulated some really important points at which one needs to take a stand on what a theory is and what an axiom is and what it is that we're looking for when we're looking for a consistency proof. A bunch of questions that come up here turn on whether or not the kind of consistency that we now look for in the mathematical theory, namely the kind of consistency that's demonstrated by giving a model, whether that kind of consistency is something like consistency in a pre-theoretic sense and also questions like whether or not the kind of entailment relation that we talk about now that we demonstrate by giving models, whether that is the same thing as a kind of pre-theoretic entailment relation. Fraga's articulation of very clear systems of logic and views about them made it possible for us to raise questions like that. And he has a very clear view, I think, which is that the answer is roughly speaking, no. In fact, there's a big gulf between the old-fashioned intuitive, compelling questions about consequence and consistency and independence and the new notions. So in any case, I think finally a large, well, an important part of Fraga's legacy is that he has made it possible to ask and to answer with some technical detail, fundamental questions about the nature of logical relations. Okay, so on a more personal note, you've been one of the best philosophers that we have, living philosophers and historian of philosophy, but what's your advice for those who want to get into professional academic philosophy? My main advice is vote for people who value state-supported education because higher education in general is in a very bad way at the moment. I was very fortunate to have been educated in a time when public universities in the United States were free or darn close to free and very, very, very good. And this was also a time when there were a reasonable number of jobs and these are connected points. This was the era in which it was thought of as an important part of society to have good universities and that it was something that we, as a community should support. And this has fallen by the wayside in many, many countries at the moment and has made it both much more difficult for people to get a university education. And unfortunately, much more difficult for people to get a university job. So there aren't nearly as many good jobs as there were. So anyway, that's my first point is advocate for a much more robust kind of support for the academy. But more specifically, yeah, how to think about an academic job. You know, the only thing to be said is people stumbled into academic jobs. They find an area of inquiry that they fall in love with and they end up going to graduate school not typically because they think I've always wanted to be a university professor or a professor of any kind, but because they just don't wanna stop thinking about their topic and hopefully because they've also discovered that they have a certain expertise at teaching and they enjoy teaching. That at least is my story. I love teaching. So I think really the thing to be said about an academic career, how to prepare for it is you've gotta follow what you love and also be aware that it might turn out that you have to lead it too. So keeping your options open, especially if you are trying to support any other people besides yourself is important. It's really hard to become an academic these days. No, is a career in philosophy worth it? Or would you say that your career is worth it? My career is worth it, no question. Yes, but I am extremely fortunate. Yes, I love my job. I have an extremely rewarding job I have. Yeah, I do what I love. I adore teaching and as painful as writing philosophy is I do like figuring out the things that I finally write as papers. And so yeah, it's a real gift to have an intellectually fulfilling career. So yeah, I mean, if your question is, is it worth it for me? Absolutely, I've enjoyed almost every step of the way. If you mean is it worth it now? I mean, I'm old enough that the job market was quite different when I looked for a job. And that's, that really is going to depend on the person. I think there are easier ways to make a living. To say there are ways that get you into making enough money to live on a lot earlier and in a more guaranteed sort of way. It's a real hit and miss kind of thing. I am under no illusions that it was merely my skills that got me a successful career. It was an enormous amount of mere good fortune and help and luck. So yeah, it's a great career to have but it's not always a great path to get to it. Okay, so thanks again Professor Blanchett for sharing your time with us. And for you guys, please like and subscribe to my YouTube channel. Joining again for another episode of Philosophy to what matters where we discussed things that matter from a philosophical point of view. Cheers. Thank you.