 Hi, welcome back to history and philosophy of science and medicine. I'm Matt Brown. Today we're talking about Emre Lakatos and his philosophy of mathematics. Emre Lakatos was born in Hungary in 1922 and he emigrated to the United Kingdom in 1956. He led a somewhat ignoble career in Hungary as a Communist first student then intellectual and as a Stalinist kind of revolutionary. His disillusionment with Stalinism came just a few years before the failed uprising against the Stalinist regime in Hungary and the subsequent crackdown by the Soviet Union and after that he fled Hungary. He completed his PhD in the philosophy of mathematics in 1959 at the University of Cambridge and within 10 years he had risen to a professorship and a major reputation in the fields of philosophy of mathematics and philosophy of science. Unfortunately, Lakatos died of a heart attack in 1974 when he was still relatively young and kind of at the height of his philosophical career. Now Lakatos encountered Karl Popper's work already when he was in Hungary some time around 1953 but before 1956 for sure and this seems to be one of the things that led him to cut ties with Stalinism and develop a different kind of philosophical view. He became a colleague actually of Popper's at the London School of Economics in 1960 and he saw a lot of his work as extending and improving upon Popper's ideas so there's a lot of relationships there. Karl Popper was notoriously difficult as a person as a sort of intellectual colleague and so Lakatos's desire for improvement of Popper's ideas ultimately led to a kind of breakdown of their relationship in some respects. Lakatos was also very good friends with Paul Fireaband, another apostate-Paperian if you will. Fireaband's most well-known work against Method which we've talked about before was originally actually conceived as the first part of a collaboration between Fireaband and Lakatos where Fireaband would launch his attack on Method, on Rationalism and then Lakatos would restate and defend a sort of fallibilistic critical form of Rationalism, a kind of pro-Method position. Unfortunately Lakatos died before he could write his response and so all we got was Fireaband's attack although you can pick up on some of the ideas from lecture notes and the correspondence between Lakatos and Fireaband which is collected in this book for and against Method. I want to talk a little bit before we get into Lakatos' philosophy of mathematics about the other idea in philosophy of science he's most well known for which is his methodology of scientific research programs which I think we can see as an attempt to kind of take the best ideas from Popper as well as ideas from Kuhn and Fireaband and synthesize them into a somewhat more moderate or rationalistic view. So it's important to think of this as a kind of modification of falsificationism right so according to Lakatos the right unit of analysis for thinking about science is not the theory but what he calls a research program and a research program kind of like a paradigm has a kind of central set of commitments Lakatos called this the hard core of the of the research program and it had what Lakatos called a protective belt a set of less central assumptions less central commitments that could be modified over time but Lakatos recognized along with Kuhn and Fireaband that every theory every hypothesis is sort of born in a with a set of anomalies right that it couldn't really address and rather than treat these as automatic falsifiers Lakatos articulated a view in which the research program could be relatively tolerant of anomalies as long as it was sort of attempting to address them over time. So there's a kind of version of Kuhnian normal science here in which what the research program does is sort of modify the ideas the theoretical commitments the postulates the methods in the protective belt in order to accommodate anomalies and increase sort of empirical content. Now another aspect of the methodology of scientific research programs is comparative right so you might have multiple research programs going on at once and it might be competing with each other under certain conditions this is sort of more like Fireaband's pluralism in a way for Lakatos their activities like normal science and like scientific revolutions going on simultaneously most of the time. Now a research program could be could be what Lakatos called degenerating or progressing right a degenerating program is one that is making sort of ad hoc changes to accommodate anomalies but not increasing its sort of predictiveness its fruitfulness its its sort of scope over time. And a or a research program could be progressing that means that it is it is it is growing it is it is covering more phenomena that were not previously explained it is making modifications in response to anomalies but those modifications increase rather than sort of decrease the predict novel predictive content of the theory over the research program and if one even more well established research program was degenerating was sort of decreasing in content was making all these ad hoc changes and another research program was progressing it was growing it was improving it was expanding its empirical content it was making more novel predictions then it would be rational according to Lakatos to sort of jump from one research program to another and in in this way you can you can see I hope how Lakatos kind of tried to capture the best of both worlds with Popper and Coon on the one hand the notions of degenerating and progressing sort of replace the sort of naive picture of falsification but they still capture the idea that if all you're doing to is changing your theories so that it sort of turns refutations into or conflicting instances anomalous instances into supports for your theory without making new any new predictions new risky predictions that was still bad right so the the role of risky novel prediction is preserved but also the Coonian fire Abindian insight that every every theory is born in a sea of anomalies is is also captured right and Lakatos argued that this was both a better descriptive account of what scientists are actually doing and was was entirely rational way to proceed um no that's not what you read about for today but I think it's important to know that he's sort of developing this sophisticated theory of science um that he sort of presents as inspired by Popper or based in Popper as an improvement on Popper but what we want to really focus on today is Lakatos's philosophy of mathematics and the work that Lakatos titled proofs and refutations in a kind of direct reference to Popper's own conjectures and refutations right so what what is I think most central for Lakatos's philosophy of mathematics is that he was opposed to formalism what he called formalism um he preferred what he called a quasi empirical approach to mathematical knowledge which we'll talk about in a moment but I think most centrally Lakatos was um critical of sort of views about the nature of mathematics and mathematical knowledge um that that focused on it as a purely formal system the number of reasons for this some of which we see unfolding in the reading but I think one of the core ones is for Lakatos understanding mathematics in a purely formal way means that there's no way to understand the growth of mathematical knowledge the discovery of mathematical knowledge in other words it totally separates the philosophy of mathematics from the history of mathematics um on the other hand Lakatos thought that that was really where the really interesting action was in our sort of understanding what math is and how we create knowledge in mathematics a formalism for Lakatos is a particularly broad notion and includes what is more narrowly referred to as formalism um which is the idea that math consists of kind of arbitrary symbols and rules for manipulating those symbols um a formalism on this narrow conception is a kind of purely syntactical um system with no no built-in meaning or interpretation other than the relationship between the symbols and and doing mathematics is kind of like playing a game right um playing a particularly abstract game another view that Lakatos encompassed with his term formalism is what's what we call logicism right this is the idea that mathematics ultimately can be reduced to formal logic that we can use the machinery of formal logic um uh to sort of derive all of the truths of mathematics um so here again basically um mathematics is characterized as a formal deductive system um and then another view uh which is known as intuitionism is also encompassed here is under Lakatos' critical scrutiny um intuitionism is the idea that mathematics is basically a kind of mental construction there's a sort of um mathematical inquiry is basically a process of constructing abstract objects in our mind um and what we do when we we posit axioms and make proofs and do all that is um is a constructive activity not discovering something outside of ourselves right in all of these views the kind of prototype um of a formalism Lakatos' broad notion of formalism is Euclidean geometry right so you may remember from you from geometry class that Euclidean geometry basically proceeds by positing certain axioms and then deducing theorems through formal proof that's sort of the nature of of mathematical knowledge now um how does that work okay so you start with axioms these are these are posits these come from um well it's it's actually not clear where they come from um but you start with with axioms about which you're certain and then you engage in deductive proofs based on those axioms and some proof rules and you derive theorems right um and uh whatever truth and meaning that exists in this system uh derives from the axioms so any particular theorem maybe it's a theorem about the relationship between angles and um sides on a triangle it's it's going to be true because the axioms are true and the meaning of the of the any concept derivative concepts in those theorems is derived from the meanings inherent in the axioms right so these are these are what um what Lakatosian one place refers to as Euclidean systems these kind of formal deductive systems Lakatos wants to contrast this uh with a different way of thinking about mathematics in which you're you're sort of broad general um statements are not so much axioms as conjectures right a variety of of conjectured or hypothesized um sort of general formal laws right there's still a place for proof right the proof helps us derive um so the deductive proof helps us derive more basic statements from those conjectures but the form of proof may not be um uh is so much a straightforward deduction is more like a thought experiment right now on this account on on Lakatosian's account here um truth doesn't so much derive in this downward direction as it goes uh up from the basic statements to the conjectures right but we know right uh that when we're talking about a deductive system um sort of basic you can't quite transmit truth that way from the particular to the general that's the problem of induction when we're talking about empirical science and it sort of applies here too but what you can transfer this is the point that popper made his career on in a way you can transfer falsity in that direction right if you derive a basic statement which you know to be false then what you have is a kind of refutation of the conjecture right um and so you can you can transfer um falsity in that direction this is what characterizes what Lakatos calls quasi empirical systems um and and crucially Lakatos thinks this is really where the development of mathematical knowledge is he uses in proofs and refutation the form of a dialogue between a teacher and students to demonstrate this sort of uh the sort of way that this quasi empirical inquiry is supposed to work right part of the criticism uh that Lakatos is engaged in when he's thinking about the idea of formalism is that it's not clear from where the axioms derive their authority that's sort of the core problem with formalism for Lakatos and and it makes sense in historical context because um you know to take take Euclid's geometry right the parallel postulate in Euclid's geometry was always a source of some concern right the parallel postulate seems like a little too um a little too uncertain to be the basis of uh of the system right and many tried to derive the fifth postulate the parallel postulate from other postulate the other four postulates right the other four axioms um it turns out you can't do it and the revolution of non Euclidean geometries that took place in the modern period was a kind of real driver for the sort of thought in philosophy of mathematics that that Lakatos is responding to now Lakatos did think there was a place for for Euclidean systems in mathematics and and but largely it was something that took over once the um the sort of inquiry into in the quasi empirical realm was done right so there's a kind of process of formalization when the sort of the unrefuted conjectures and their proofs had been settled on where you would kind of turn mathematical knowledge into a formalized system and there's a kind of like activity like normal science in a way for Lakatos which you might call normal normal mathematics right um uh where there is some puzzle solving in a Euclidean framework some attempt to clean up and make more secure the proofs for various theorems but by that point according to Lakatos the kind of significant work of the sort of significant creative work of the mathematician had already been done and the real sort of locus of uh knowledge creation was here in the quasi empirical systems so that's a quick introduction to Lakatos's philosophy of mathematics I look forward to talking in detail with you about Lakatos's criticisms of formalism and the dialogue he presents in proofs and refutations and I look forward to hearing your questions you can you can leave a comment or write on to the discord or I will see you in class um uh otherwise I will see you next time