Speed-Up, indispensability & unfeasability

0:00 and red brown speech subject I will thank Phyllis and Volker for inviting me to this talk. While the talk is about the topic I can sum it up in two sentences. The first is the mass and the speeder over the naughty woman, excuse me. Does this create a problem? There's a handout which should come round. There's some gum. which I apologise for all the gunk, you're the gunker. Now there's some initial stuff that I'm not going to ignore, but we might come back to it. Kind of philosophical stuff about motivation, phenomenalism and ordeals. Which one? speed up in this talk. So it's then called the Indispensibility Argument, which is famously associated with Quine Petman, and also, in fact, an unpublished paper by Goebbels from 1953-59 called his mathematics logical syntax of language in which he also develops a kind of version of

2:30 to be the Dispensibility Argument. This Dispensibility Argument says mathematics is, our scientific theories are thoroughly mathematicised, they, mathematics is necessary to formulate our scientific theories, our theories quantify over real numbers, natural numbers, every part of various mathematical structures and stuff. We have laws of physics, I don't think it's one, down square by phi, And obviously phi and rho are functions from the space side. So, the argument is that our theories are mathematicised, they're quantified as mathematical objects. If you're committed to our theories, if we think they're true, even if we think they're just approximately true, then you must be committed to the mathematical entities that you quantify in our mathematical theories. Furthermore, there doesn't seem to be an easy way of getting rid of commitment to mathematical entities, so therefore it's an argument against . Now, the usual administrations of indispensability involve rather abstract mathematics and theoretical physics. The use of topological math or R4 to describe space-time, things called representation which shows how you can take a system of non-ronistic actions and show that they can be any model of both atoms to be represented in a mathematical model, and states of physical quantities which are understood as functions from physical domains from mathematical domains. I want to discuss a simpler example of mathematics where we apply mathematics to problems of logical validity. So they don't, the examples are not as abstract as the usual examples that are discussed in this literature. The pen's book by Hartree Fields, in fact, I have the certificate of numbers, which is the pen's book by Hartree Fields, attempting to show how to eliminate quantification over real numbers and so on and so forth from scientific fields. Now, one of the things that I want to, one of the examples I want to mention is this kind of example here, where we say that the number of f's is n. This is something that nominalists often mention as a kind of triumph, because instead of saying the number of things are n, we can say there is an x1, there's an x2, there's an x3, there's an xn.

5:00 They're all different, and they're all x. And furthermore, anything with an f is one of those. and that's meant to be a success because you've eliminated reference to the number and you're now just going to quantify as a ranger over your concrete, you know, the number of elephants in the zoo is seven. You're now referring to the seven elephants. You're not referring to the number seven as well. But actually I would argue that this is a failure. This occurred to me about a year ago that I was thinking about some simple examples of the sort of thing that you sometimes say in logic lectures that you can do 2 plus 2 equals 4, you can do that in first order logic, but of course you don't want to set that in a logic exam. I mean, if you just try and work these out, they're just horrible, they just go on, and I think I worked out that one took about 27 lines, you know, once they go up they get more and more horrible. The topic connected to this is speed-up, because speed-up is a phenomenon which involves the idea that one theory can prove things a lot more quickly than another theory. And the idea here is that mathematics can actually prove certain validity facts much more quickly than the original fact itself can be proved using a standard deductive system of first-order logic. The general topic of the applicability of mathematics to problems of logical validity has not really been widely discussed in this literature on indispensability. It's been discussed in a paper for our country field called, Is Mathematical Knowledge Just Logical Knowledge? And there have been a couple of papers by George Bulos concerning Speeder where he discusses kind of technical issues. to the indispensability arguments or discussions about normalism or anything like that. There's a short paper that, in fact, a short summary of this talk is going to appear in analysis this month, and there's a paper on John Burgess' website, for instance, called Protocol Sensitive to the Light Logicism, which contains at least some similar ideas

7:30 This is the example Boulos gave. You have the following inference. We have three formers here expressing some constraints on function f. That's a kind of Ackermann-like function. It's not a standard Ackermann function, but there are a number of different versions. And then this says that the number 1 is in the extension of d, 1 is a d, and if x is a d, then the successor of x is a d. And then this says that the f of 5, 5 is a d as well. People also point out that 6 is actually a logical consequence of the premises 1 to 5. We just need to work out the value of f of these two arguments. But unfortunately, and of course he used this thing to step through, but unfortunately the value of this function is huge, tremendously huge. He points out that Schultz's first order derivation has length at least an exponential stack of 65,500 versus six twos. But there's a one-page derivation of the conclusion six from the premises using second order comprehension. So if you're allowed to use second order logic, or if you're allowed to use set theory, you can roughly assert the existence of the closure of one under the successor operation, do various steps, show that the function is... Sorry, those are the only axioms here. It's not, it's not, these are just logical. Yeah, yeah. The main idea is you've got a function here that grows very rapidly, and then you've got a... I see what the idea is, but what you're saying, this isn't in arithmetic, this is just logic. Pure logic. Pure logic. Pure logic, yeah. Okay. And in fact, there's a one-page derivation of this. In fact, Boulos has the one-page derivation. I call this formula Bill Lotz 4-4, and it's just this formula, the paper formula. I like to think of A as zero, whereas he thinks that A as one, it doesn't mean it's a critical difference. There's the formula, it's a valid formula, this formula is valid, so it has a proof in the system of personal logic, but you don't want to give it to one of your logic students in an exam, unless they know some second order stuff.

10:00 Prove that the following Thornry is valid using any method you can think of. OK, this phenomenon is called speedup, and it was first noted by Gödel without proof, in fact, in 1936. And it concerned the speedup of n plus one, n plus one arithmetic over n order, n plus one order arithmetic over n order arithmetic. This is a statement to the results. Ready recursive function gene is an infinite class of formulas. They can be chosen to be pi-1 formulas. So if k is the shortest group of phi in Zn, and L is the language shortest group in Zn plus 1, then k is greater than G of L. So I kind of indicate it like this. Zn plus 1. We have a proof of that K right here. We have a proof of... I'm just going to wait a minute. So this K proof is one of this. And then K is going to be greater than G of L, so this is going to be a much longer group in the weakest end-order order of the two. So, and this can be some recursive function, so it could be, you know, an exponential stack of two is L high. Right, so it could, so, I'm kind of going to hate it. Goethe wrote, thus passing to the logic of the next hyper-order has been valid, not only of making provable certain propositions that were not provable before, but also of making it possible to shorten by an extraordinary amount, infinitely many of the proofs are already available. And Bulos comments on his own example. The fact that we so readily recognize the validity of I, that's his inference, would seem to provide as strong a proof as could be asked for that no standard first of the logical system can be taken to be a satisfactory idealization of the psychological mechanisms or processes, whatever they might be, whereby we recognize first or the logical consequences. So Billas' conclusion actually concerns psychology concerns, something to do with the way that

12:30 the mind recognises consequence, whereas I'm going to draw a conclusion about epistemology, about what would justify the belief in to be true. So you may come away from this talk and think, oh, Billas is right, but Kepler is completely wrong about the list, if you disagree with the epistemology. The first thing I've started thinking about was the complexity of statements of the number of Fs is N. To be expressed normalistically like this, that's a way of expressing that there are at least N Fs, and that's the way of expressing there are at most N Fs. Now what you want to prove in some kind of reduction, some kind of analysis, if you want to prove the statements that you have by conditionals of this form, which I call end sentences. The x's are a fair line difference? Well, in here? Oh, yes, I'm assuming that the... In fact, actually, I'm thinking of these syntactically as x concatenated with a 1. Oh, okay. And then this is x concatenated. The way I think of this is actually going to be a numeral, a binary numeral. It's actually going to be a simple complexity. In fact, the first time I did it, I was thinking of raw x, raw x prime, raw x double prime. So I was using a kind of unary notation. I mean simply that xi is not xy, right? So now the value is . As long as the model, as long as you admit of enough things in the foralls to range over, you're quantifying over enough terms so that you have at least n things and then I don't think you'd need to be saying the disjunction Ted actually mentions these examples actually as a condition on the adequacy of the definitions the number zero and successor operations and so on in favor of the that calls this convention M and says that their analogies to task is T-sentences. And in fact, these come up in the example that Hartley Field gives of how mathematics

15:00 is actually used to simplify our reasoning about normalistic facts. So this could be the number of elephants in the zoo is seven, and then it could be expressed normalistically like this, or the other way around. I think roughly the idea is that we express it nominalistically, then we express it using numbers, then we reason using numbers, and then we translate back into a nominalistic way of expressing it. Simple arithmetic facts of the form n for n equals k are expressed like this. There are n f's, there are n g's, there are zero things like f and g's, therefore there are k, things like either f or g, and these obviously come out as logical truths. do this? Well, the complexity of these statements with definite cardinality, numerically definite cardinality quantifies is, I estimate a constant times n times log beta ten n, where this ten refers to the way that we express the subscripts on the variables. If you express them using the subscripts using binary, then it would be love base 2. So the capacity of a term in base N, which goes by long N. Then you get roughly N conjuncts, so it's roughly N. So the complexity of this statement, there are at least one million books in Cambridge University Libraries, approximately ten to the seven symbols, so if a nominalist were to write that down, it would require a book about 3,000 pages long. So if a nominalist really believed that they could actually write out this statement without using the expression referring to one billion, then they would need a 3,000 page book. I'm assuming 3,000 symbols per page, which seems to be that best to be that right. Yes. S and G are supposed to be specific predicate constants? Yes, they're just arbitrary predicate symbols. Yes, but then if you say it's re-expressed as, then it should actually be an infinite

17:30 where you instantiate f and g by all the predicates that you have? No, no. When I say re-express that? Here. No, no. I'm not. You say n plus m equals k, re-express l. But that's a specific sentence concerning f and g. Yes, right. Whereas n plus m equals k. Sometimes it's written with two prefixed universal second order. Yeah, that would be the re-express. But then you have a second order, right? Yes, yes, yes. I'm not sure whether that's in our crucial importance, but, yeah, you could do that as well. In fact, Anne will be talking about a load of ways to express these statements, and as formulas, they do actually come out as either logical schemas, schemato, or they come out as second-order formulas. And here's the example of, Boulos' paper is called Some Curious Imphases, and the examples that I'm giving are examples involving arithmetic reasoning, which don't grow anywhere near as fast as Boulos' examples, but they grow roughly like n squared, so they're reasonably, you know, if you were taking an example of, say, where the parameter is 100, then it's going to be something like 10,000 or something steps in all. So we can express, re-express an arithmetic sum of this. Either a logical schema like this, or as Hannah said, by prefixing this with second-order quantifiers. There's a second-order formula. Obviously, this formula of sum N, M, K is valid if K is the sum of the two parameters, M and N. The calculations grow polynomially in the time of K, and a number of sentences is like K-cubes along with K. So for example, well, I just worked out this example using system of natural deductions, and it required about, so that would be, that's a proof of, if there are at least two As and at least two Bs, then there are at least four things that are either A or B required about 57 lines. We can also re-express products like this, actually I I was thinking about this yesterday, but I'd actually made a mistake here. This should be the other way around, around the X's and Y's. But again, you can express products as logical schematra as well.

20:00 And the product is valid in a case . Here's another example. It's the example that I put on the abstract. It's the 101 Dalmatian example. I actually spelled this incorrectly when I sent it to Volker. Which did you misspell? Dalmatian. Not foul. One that hurt. Food. We're going to put foul. And for this inference PHP 100, PHP stands for Pitch and Whole Principle. We have, there are 101 Dalmatians and there are 100 food bowls, and each Dalmatian uses exactly one food bowl, so we'll talk here. And obviously, if each Dalmatian uses exactly one of the people, then at least two of the Dalmatians must be sharing one of the food clubs. Now, obviously using mathematics would prove that this is valid in a few lines, and it's the kind of undergraduate, tardy trick that undergraduate mathematicians do, when they have, you know, more than a certain number of people, they say, at least two people in this room were born on the same day of the month, or something like that. I have more questions. Shouldn't nine have to be responsible? Shouldn't. Because seven, eight are nine. Together are two sisters. Are they? Yeah. Yeah. Oh, should it be at least one? Of course. Because if they use exactly one, it's one bowl. We don't need two bowls. At least two bowls. No, no, they shouldn't be. Yes, yes, yes. I just want it to be a function, that's all. Yeah, that's right. I'm getting the same answer. Yes, sir. Oh, you do. He's right. He said you can't read that one. Anyway, we've proved this using the Fischer Law principle, which is that if x and y sets,

22:30 and the carminality of x is greater than the carminality of y is greater than 0, and f is an arbitrary function from x to y, then f must be non-injective. But the shortest first order derivation of 10 from 7 to 9 in a nominalistic formulation, these are rough estimates. I estimate to be around 10 to the 7th symbols, which is roughly this cubed times a constant. The number of steps in the inferences is roughly n-squared, and the simple complexity of proof is roughly n-cubed. So, fully written out, the first order derivation for this Dalmatian 101 Dalmatian inference would again fill about 3,000 pages, I estimate. So, for a human being writing a page in 30 minutes, it's about six days' work, non-stop. I'm sure somebody will come up with an objection saying that it would really speed this up by some fancy trick. Another example, I won't give a few examples, I'll give the Milne example, which Peter Milne gave you. Everyone has at most two parents, so everyone has at most two to the end, end parents, where an end parent, a one parent is your parent, a two parent is your grandparent, A three-parent is a great-grandparent. And it's clear that that's valid, and if you might say it's valid. Formalizing like that, there are almost two to the n, y, such that there's roughly a chain linked by the relation r. The proof intuitively just involves a tree, a binary tree. Again, the length of the derivation here is exponential because we just built it in here. So, this inference here, everyone has at most two parents, so everyone has at most 32 great-great-great-grandparents would need around 40 pages, I estimate. And you can get them to go quicker and quicker. But these are meant to be inferences that are meant to be obvious. They're meant to be kind of intuitive. of your own. It's rather awkward giving these examples to mathematicians because you have rather finely honed intuitions about what's valid and what's not valid. But even in some other cases where you actually give them to people who aren't trained mathematicians, they'll say, oh, that's obviously valid. Whereas the Boulos example is not just completely obviously valid. I gave a quick sketch here of the mathematical proof. The mathematical

25:00 In any model of the expression in square brackets, you have n plus 1, a's, and you have n, b's, and f is a function from the a's to the b's, so therefore, by the fictional principle, the function is non-injective. So it's valid for all n, in fact, and by differentiation it's valid for 100. So the mathematical proof is not really long, just by inspection. But if we reject as false the mathematical premises, we can't detach the conclusion that PHP 100 is valid. And the nominalist does reject these premises as false. That's the algorithm I'm given. So, hence the anomalous needs to show how to establish the validity of an inference, the validity of a formula, an understanding as either a second order formula or just as a logical schema, without using the mathematics. Or they need to give some kind of account of why we can use the mathematics, even though we think the mathematics is false. I'll skip over this, which is just a whole load of formalizations of those things. The mathematics used to show that these valid, I call them quasi-arithmetic formulas, they're just really ways of coding arithmetic statements into first order credit code modules. The mathematics used to show that these formula for values is just simple combinatorics, while Bumas' example turns on these rapidly growing recursive functions, such as the echo function. Non-mathematicians I speak to tell me that the pigeon-hole inference is obvious, and the other ones are both to be easily explained. In a philosophy and math seminar I gave about a month ago, a student with very little form of mathematics background even predicted the conclusion for this inference that the number of people in the room is four. I hope I haven't got an inconsistency here. The number of people in the room is four, and the number of houses in the streets is three, and each person in the room isn't exactly one house in the street. She said, oh, at least two people must live in the same house. And when I gave this talk about a month ago, somebody did ask a question about the research that has been done about psychological research on logical inference and how it all turns out that our ability to carry out logical inferences is rather context dependent.

27:30 So if the predicates refer to concepts or involve concepts that we immediately understand that we seem to be better at carrying out that kind of reason. Did she already know it, but not know that she knew it? It's a violation of the K.K. Principle. By it, I mean the Pitchcock Principle. And related to this is Plato's meaning. Perhaps we possess some mathematical knowledge prior to instruction, which is innate, in some sense, and it's just brought out when we engage in instruction as a teacher. Doesn't the problem get harder when the numbers get bigger? Well, we're three and four. You can sort of imagine those three kinds. Yeah, well, this will come to one of the responses. This is one of the responses is, are we actually performing the computation, imagining it in our head? Or are we seeing the general results somehow abstractly? And that's part of the argument. I agree that if you just have got two, you've got three Dalmatians. and two balls, then obviously the combinatorial possibilities are very small, but we do seem to be able to generally see that no matter how many balls we have and how many Dalmatians we have, as long as there are more Dalmatians than balls, then there must be two, any two of these lengths must converge. We do seem to be able to see that as a general fact. I'll come back to that Here's another example that I think I'll talk a little bit about It involves the zeta function but it's a really trivial example based on the zeta function We know that zeta of 2 is pi squared over 6 The novelist takes this equation to be false because they don't believe in pi or numbers or zeta functions or anything like that But these are computable reels so there should be two intentionally distinct programs the n-th decimal digit of zeta 2, and the second calculates the n-th decimal digit of pi squared over 6, using some standard definition of pi, which is not the same definition that you started with. From that one, we can prove that for all n, the c1 of n is equal to c2 of n. I wonder whether that's proved properly here. It should be proved, I mean, there should be some obvious arguments, because this is a recursive function, so it should be representable

30:00 It should obviously, I mean, I don't know whether this is generally true, but there should be some obvious argument that recursive equations between recursive reals should be, I don't know. I was hoping that maybe one of you techians might be able to answer that question. It's improbable. Improbable, but it should be critical. In general, well, in this case, it's called mutual. There's a general insight that will be some older participation. Yes. It might depend upon whether it is provably recursive functions or something like that. Ah, yeah. It seems to be on the proof, on the inspection of the proof. Yeah. Anyway, suppose that we have two physical systems actually physically on the table. One of them exemplifies computation one, the other one exemplifies computation two. Then we can make the following empirical prediction. It's an empirical prediction. If P1 is started with a token of M, and physical computer 2 is started with a token of M2, then they will eventually compute the same result. That's an empirical prediction, although it's derived ultimately from this mathematical equation. And it seems that the Assumption 26 is actually necessary to actually prove it. Of course, if this is provable in PA, then we'd only need PA. It wouldn't need the Fourier analysis or whatever that was used to prove this. So how can a nominalist explain the truth of this prediction? I mean, if this actually were a prediction, we actually did have the two computers in front of us. One was an exemplification of this program, another was an exemplification of this program, and we put in, say, the value phi, and we just pressed a button. Can I ask a simple question? I don't really know how much of a nominalism, but do the nominists believe that zeta 2 and pi squared over 6, provided, of course, you're not defining pi pi as the same series that you're using to define the zeta, that these things are well-defined entities at all? They think that they're well-defined within the theory, yes, but they don't believe the theory, so they think that they're rather like fictional entities. Maybe I'll come back to this in the discussion about the meaning of nominism. The way that it's often described nowadays is that nominism is rather like fiction. So Pi is meant to be like Hermosimpsons. And we can sort of zoom in on him in a cartoon and know more about what he looks like.

32:30 Yeah, but there's no such thing really as Homer Simpson. All there is is the cartoon. And that's roughly the view of contemporary novelists. But anyway, Field, I think, would insist on maybe on some kind of nominalistic physics, so we'd have to give some kind of nominalistic account of the physics of these physical systems rather than saying that they exemplify this computation, and that they exemplify this computation, which are abstract ways of describing the physical system. So the point here is system actually allows you to make physical predictions that are purely physical description of the physical system in field sense, actually not referring to any numbers or anything like that, would have a problem. Okay, let's go ahead and take it now. The usual argument stated that mathematics is indispensable for applications in science involve abstract mathematics from theoretical physics. Here we've simplified the problem to one concerning the application of a small portion of finite set theory to problems of logical validity. Remember that anomalous doesn't even believe in finite set theory. In the present case, we have utilised the fact that mathematics has to feel it as a logic. So these, we have these logically valid formulas and we can prove that they're valid using mathematics very quickly, but the first order derivations might be 10 pages, 40 pages, 3,000 pages, or in Fulost's example, 2 to the 2 or two, and so on. This means that for certain purposes, mathematics is in practice indispensable for verifying certain claims, e.g. that a certain kind of inference is valid. So what is the significance of this point? So here's the main claim. Postulating numbers, sets, and functions is practically indispensable for coming to know that certain inferences are valid. To summarise again, there are quasi-arithmetic validity of such that using mathematics, one can quickly show that the formula phi is valid, using some principle concerning numbers, sets, or functions. But a direct logical verification of phi would take a feasibly huge amount of time. I'm counting even, you know, 3,000 pages of some pieces. There are technical definitions of feasibility

35:00 So the question is, can a nominalist, somebody who thinks that pi is right over Simpson, give a nominalistically acceptable reason for thinking that these inferences are actually valid? Because these are actually true statements. When I say PHP 100 is valid, that's actually true. And if you were actually to sit down, I mean, if you were actually to do it, would actually produce a concrete token which would validate the inputs. I have a couple of isolated, I think, four responses. One is contrary to appearances, we don't know that the Dalmatian example is valid. Actually, this is what Thomas Forster suggested to me. We don't know until we've actually checked it. That's like the name. So, until their computer verifies it, we just don't know. We should be agnostic about whether it's actually valid in the sense of actually writing it down. There's a professor of mathematics at Fruins College called Brian Davies, who I think has a similar view, a very strict formatist, who says that he checks simple trigonometric identities on his calculator to make sure that they're valid. actually there is a quine problem problem there actually anyway that's kind of head in the sand response I think second is something that that John maybe suggested that we actually do At least for the small parameters, we can actually imagine it and almost perform the computation in our head. We do know that the Dalmatian example is valid, but we know this biological derivation, the first-order logical derivation, which is very, very quick. And again, that seems to me impossible. So we're left with two other responses if you want us to remain anomalous. One, we do know that the Dalmatian example is valid. We don't know it by logical derivation. We know it by some other way which is non-normalistically acceptable. corresponds to what I earlier in the second time I missed, called the Hermeneutic and revolutionary nominalism. So that's a third response, which is that there's a better way of obtaining these conclusions,

37:30 perhaps by reinterpreting the mathematics in some way, maybe by some modal reinterpretation, or maybe by some myriological reinterpretation, or maybe by using second-order logic interpreted using plural quantification, and there are lots of things that have been proposed. And the final is the instrumentalist response, which is just... The world just behaves as if mathematics is true, but it isn't true. So we know that it's valid, we know it by using mathematical reasoning, there isn't any mathematics alternative, all of that is a waste of time, don't even bother with the modality, the meriology, and that's consistent with non-realism, and leave me alone. The world just behaves as if mathematics is true. This trichotomy of nominalism was actually introduced by John Burgess in a paper in 1963 called Hermeneutic. Nomalism is the idea that you reinterpret mathematics. Revolutionary nominalism corresponds to the idea that you replace mathematics with nominalistic physics. And instrumentalist nominalism is just the idea that mathematics is a useful instrument, but you don't have any obligation to explain why it's a useful instrument. It just is. We don't get worried. It seems to me that nominal is only need to ask me to try and do that from my picture, but it needs to be adequate. Yeah, exactly, yes. Well, yeah, that's the reflection principle I'm going to come onto in a moment. What they need to show is, in fact, I want to bring out an analogy with Van Prysen's empirical adequacy. So it's analogous to Van Brassen's idea of improbable algorithms. Responses 1 and 2 seem to me to be asked a question. Response 1 was, you just say, we don't know it's valid until we've checked it. Response 2 said, we're performing a very, very quick calculation. We do have reasons to think that these inferences are valid. Now, possibly there is another way to see the validity of these inferences, but the odyssey proof is probably, is surely on the nominalists, to present this other way of carrying out these proofs to show that it's nominalistically okay, one possibility involves using the same mathematics, but you reinterpret it in some way, perhaps modally. So, for example, Daniel Nolan at St Andrews suggested that you use second-order logic with a plural interpretation. Maybe I'll come back to that if people are

40:00 Doesn't that fall foul of your initial system where you can go up to say you'll need a third-order system to prove something that would be trivial in a third-order system? Yes, that's right. That might be a problem, actually. So, yes, that might be a problem with that approach. But also when you're reasoning about numbers as well, you'd need second-order logic over numbers. So that would cause a problem with that approach as well. Anyway, that's one possibility. Here's a kind of summary of the situation. Oh, this is the wrong one. The instrumentalist position seems to involve some kind of epistemological miracle. The mathematics used to show the validity of the fictional influence, the Hume influence is false, or it's fictional, or it lacks a truth value, or we should be agnostic about it, and so on and so forth. Still, the false mathematics implies true conclusions. Well, that's meant to bring out the miraculous aspect of it. How does it have this correct content, even though, at a substantial level, it's false? This seems like an epistemological miracle. The validity claims implied by mathematics are indeed facts, at least the ones that we check are facts, you know, if we check the pigeonhole inference for small values of the parameter and put them into our theorem prover, I'm sure they'll work, you know. So the world behaves as if mathematics is true, I should say true at the level of humility facts, even though according to nominism the mathematics is false and that seems to be And this is analogous, and in fact I claim that it's actually odder than the metaphysical oddity of mathematical entities themselves. The major argument against the reality of numbers is that numbers are abstract, so therefore it's difficult to see how we can know about them. They seem to be metaphysically spooky. They're not concrete things like this pen or like a chair that you can sit on or a table you can look at. They're abstract, so how can we know about them? So they seem to be both ontologically spooky, because they don't have a location in space or time or anything like that. They also seem to be epistemologically spooky. How could we know about my final objects, even if they did exist?

42:30 I'm claiming that this oddity is actually odder than this. Right, so it's even odd. The numbers are abstract, so it's very odd that we could know that. So, this is the kind of competition between the oddities. This oddity is even more odd than the metaphysical oddity of the mathematical entities themselves. And this can be compared to the belief that, say, Maxwell's equations are empirically adequate. but they give you the correct observations. If you make predictions based on maximized equations, you get the iron firing to come out the right way, and so on and so forth. But the corresponding instrumentalist position in the philosophy of science says that B is just a fiction. We don't have any reason to believe that B exists, and there's an intended analogy here between what's called a no-miracles argument in the philosophy of science, between this no-miracles argument and the standard no-miracles argument. And in fact, actually, there's a quotation that's often given from Putnam, from Murchison to Clyde, in which he says that the positive argument for scientific realism is that it's the only philosophy that doesn't make the success of science a miracle. But just shortly after that, he says, I believe that the positive argument for realism has an analogue in the case of mathematical realism. Here, too, I believe that realism is the only philosophy that doesn't make the success of the science, i.e. mathematics, a miracle. So if you're swayed by the numerical argument in the philosophy of science, you should believe in the magnetic field and in atoms and in molecules, because I know Hannah's, you can talk about it, we'll publish it out there later, but if you are swayed by the numerical argument in the philosophy of science, then you might be inclined to be swayed by this kind of New Miracles argument in relation to what's the Atlantis. Finally, I want to come on to the reflection principle that I mentioned briefly. Actually, I did have another

45:00 overhead which actually had a short discussion statement to this, but I managed not to print out, so I'll put that here on the last one. By a specific proof, everybody agrees, the nominalist, everybody agrees, the nominalist, the extractivist, the realist, agrees that mathematics proves that phi is valid. You have that right in front of you, it's just a concrete thing that you actually have. Now, if the mathematics is true, then we have that if the maths implies that phi is valid, then phi is valid, and that's a reflection. Like Shapiro's guru, the guru says, P, then P. On this, we can infer that the corresponding instance of the vision hall inference is valid. And we can also infer that it can't be feasibly proved in the appropriate sense of feasible. I mean, you know, not without doing it on a super computer, taking several days or something. Oh, in actual fact, I'll bring that up to how we can take that. I explained that a proof of this instance of the Pigeon Hall Principle has 10 to the 19 symbols, and that it would require about 100 million years. So this is DHP 1 million. Now, I've actually got a bit of dialectic here. related to this issue here about whether the nominist is entitled to believe this reflection principle which I think is a crucial issue whether the nominist is entitled to believe this Hartree, representing the nominist, believes A but does not believe B, C or D so has no reason to believe that this instance of the pigeonhole influence is valid Hartree will never in practice deduce that this instance is valid. Hartree of course can note that Hilary thinks, Hilary is Hilary Putnam, thinks that Hartree could in principle deduce the validity of this, taking just over, whatever it is, 100 million years. Hilary replies that although he agrees that a near immortal Hartree anomalist could in principle deduce the validity of this inference in around 100 million years. to suppose that he could, in principle, deduce this, because the only way he could infer that he could do it would be to actually use the mathematics,

47:30 which would tell him that he could do it. Well, he doesn't believe the mathematics, which does tell him that he could do it, whereas Hillary does believe the mathematics, which tells Hillary that Hartree could do it if he were almost immortal. And in any case, the very mortal Hartree couldn't do this in practice, since 100 million years is rather long. So, and even worse, Hartree himself has no reason to suppose that he couldn't either. So he has no reason to suppose that either he could or he couldn't. I just want to go back at the end, and then I'll stop, to this reflection principle. I did have a previous slide, but I managed to not copy it. This says that if the mathematics implies the validity of a certain formula, then that formula actually is valid. can the nominalist justify this now obviously the mathematical realist can justify it because they believe that the mathematics is true and that's just a reflection principle which just follows from the truth of the mathematics so the question is can the nominalist justify this, notice that this being a reflection principle actually implies the consistency of the mathematics so in actual fact the nominalists will be committed to a rather strong claim if they do claim to be able to justify this statement. If this was a set theory, then the z implies far is valid, implies valid, and that implies the consistency of z. So the novelists would be committed to a rather strong claim about the consistency of set theory. There is a discussion of something similar to this in Hartfield's paper that I mentioned at the start of the talk. In Thiel's paper, he discusses principles, reflection principles that are kind of similar to this. In fact, it's a bit more complicated. He wants to replace claims without validity by modal claims. So instead of saying, phi is valid, you say, it's necessary that phi, something like that. His argument, so far as I understand it, ultimately just comes down to the idea of... Well, I'm probably misrepresenting Thiel a little bit, but I didn't really fully understand. what his justification was for these principles. But as far as I can tell, it just seems to be that we have inductive reasons for believing in the consistency of that matters. So nobody has ever, in fact, deduced an inconsistency in set theory.

50:00 Nobody has ever, in fact, deduced an inconsistency in piano or rhythm to it. We have inductive reasons, and these inductive reasons, therefore, can be applied to justify that. I think that's Fields' argument for the justification of this. So we've got inductive reasons for believing Golbox injection, too. Yeah. Yeah. Okay. Um, and then that issue becomes a rather subtle issue about one. in fact even whether that in fact what you might do actually is in fact just say well we can look at the instances of this and say that we have inductive reasons for believing this so just cut out the middle man as it were and say that we have we've proved phi is valid in mathematics and we've checked its validity and then we've proved another phi and checked its validity and it's come out valid each time so therefore we have inductive reasons for believing in this reflection. Anyway, I'll stop there and book two questions. Yeah, I think it's a full soft line of technical words about what you're saying. Maybe I'll start with a full soft one. I mean, my, the way I read that, Ben Frost, and the whole notion of empirical allegacy is meant to be short-circuit in a miracle's argument, because empirical allegacy is true. It's true to the empirical substructure of the theory, so there's no miracle at all. That's the whole point of Ben Frost's view. It exactly doesn't make a miracle of the empirical success, because if the theory's empiricatic, then awkward work, it's empirical success. So now, if we carry that over by analogy here. I take it that the analogist's move would be to say that mathematics is true insofar as what it predicts about which fives are valid. There's no miracle there. So, I mean, there's sort of this slippage. You're saying, well, the anomalist has to say it's false. Therefore, both will be agnostic. Well, no, but I just want to reject it. They don't. I mean, all they can say is true insofar. I mean, all that matters is truth about the

52:30 The rest of the stuff, if you want to say that in order for them to believe that math was truthful, stop, and they'd have that monological commitment to do that. I'm willing to allow you to saddle them to that, although I guess one could project with that. But the whole point of doing a better process kind of strategy is you grant that, and still there's no merit. It is true. So I just think there's some solution there. another clip. So now a technical... Can I just say real quickly, but of course, all of the scientific literature objects say that it is a miracle. They just say, no, just claim that scientific theory is empirically adequate, even though we have no reason to believe in the content of the trans-empirical. That just is what we mean by a miracle. But that's silly. I mean, look, what's the best explanation? Yeah, I know. But, I mean, that's, it's a fallacious form. I mean, it doesn't, you know, I mean, it's supposed to be the empirical success. That's the explanation. That's not miraculous. In fact, that's predicted. The only way that the scientific theory, the realist version of it, predicts the empirical accuracy is via the empirical accuracy of the theory. It's via the truth of the empirical substructure. So, if there's, I mean, there's, and all that you're doing by being a realist is you're adhering to a less probable theory theory with more commitments. I mean, so I don't, I mean, that's a really, there's a, you know, I mean, just to be careful about Frosty, I mean, I think that was the whole point of the strategy was to circumvent this miracle. But anyway, my technical worry is, you have all this stuff about, in other words, there seems to be another, technically, another option that you have to consider, which is, it is a logical inference, okay? And, but the way you could do it would be this. I mean, you seem to be assuming that there has to be some general purpose, strategy for doing the difference. So you're assuming some kind of, right, complexity is always relative to some kind of proof system. Right? So this notion of complexity, full stop, I don't know what that means, right? I mean, so take just from calculators. There are some theorems on certain expectations that have really long periods. So, why couldn't it be that people just have lots of different ways of reference, lots of different logical systems, and they're very good at picking out the right ones in the context that they have a short course for that problem? So, in other words, there's another option you have to consider, which could be that

55:00 That wasn't enumerated in your list before. Well, how would that apply to this, the kind of view, the original view of the speed of the sample? You've got a way of exercising logic such that that theorem really is quick in that system. Well, it's a short proof. Right. Now, I agree that there will be a low toxicity to the way that you formalize the exercises. Wouldn't there be miracles then to show that all of those logics proved consistent, true things? Because you'd have to have some reason to assume that those logics were themselves correct. that pushed the problem up to that level. Some of them are going to be bad, though, as well as some of them being good at specific things, some of them are going to be bad at specific things as well. And the nice thing about going beyond it to the mathematical reasoning, which doesn't just reside in some kind of recursive of actualization of logic, is that it's going to have a kind of generality, because the examples that I'm discussing involve general combinatorial principles about numbers and functions and sets. So, I agree that there is a certain amount of context sensitivity in the formulation of your automatic system, but I think that, again, I'm not sure if that response is going to work, because there are only some cases where it's bad, whereas the mathematics system would be generically better, something like that. And on the... That would be more responsive to the technical issue. I mean, one could just take some of these valid inferences and take them as actions in your logical system, and there would be nothing wrong from the point of view of logical methodology, I think, in doing that. but there would be something slightly dodgy about taking the validity of all the pigeonhole implements. On the philosophy of science issue, I think what I wanted to do was to draw out the analogy between the two arguments, rather than actually settle the issue, just rather than draw out the analogy between the two, the instrumentalist response in the... He hasn't really done that because if you do that, then this whole question about how they're entitled to reflection, they get it for free, that's not an issue. Because M. Frostman gets for free truth. He gets truth of the empirical sub-structure. Well, I mean, they can get truth of the part of mathematics that's entailing the validity.

57:30 So I don't see why there's a question of their entitlement. I mean, you have to be more careful about what you don't like, what they're not entitled to, and what role it plays in predicting those validity. The fact that they're numbers, that doesn't matter. That's not part of what gets you those beliefs. Just like in the realism case, it's not that they're electrons. It's not the fact that they're electrons that gets you this high-level inference about science being empirically adequate. Science being successful if it's empirically adequate. Whether they're electrons or not is moved to that prediction. So div B equals zero just happens, just happens to predict. Predict this pattern. You don't have it. Fine filings. Even though we have no reason to believe B exists. That's the miracle. Or zero, come to that. Jeff, Jeff, you can make your argument stronger here. I normally don't mention Van Frasi, but just to help you out here. The explanandum in the sophisticated versions of the No Miracles argument isn't the mere empirical success of the theory. People like Boyd argue that the explanandum is the success of theory-laden methods in the history of science. So they think it's not the mere fact that this particular theory is empirically adequate. It's that when we use background theories, which themselves refer to unobservable structures, to generate new theories, consistently over time, theory-laden methodology like that, using background theories that refer to unobservables to calibrate instruments that you... You know, new domains, it's all that stuff, the theory-ladenness of the methodology of science, which consistently turns up new, predictably successful theories. That was supposed to be the explanandum for the more sophisticated version of the No Miracles document, than just saying, OK, we predict where the Iron Files is going. Sure, I agree, but it's just a generalisation of that. It's just that with spades on, basically. It is that with spades on, but you can say a little bit more in response, I think. So you'd want to say something analogous about the history of mathematics. The theory-ladenness of mathematical methods in the history of mathematics when it comes to finding out about new and deeper mathematical structures, which then give you further valid, you know, from them you can prove valid inferences about things that are ordinary factual matters. Yeah. Yeah. So that's one of the issues.

1:00:00 I don't think the issue is whether they're entitled to reflection. I don't think that's really an issue. That's not analogous to what's really an issue. But in the case, in the case of the scientific case, what we, as I take it, what happens in the scientific case, on Van Prassen's view, is you've got to say, you know, the magnet, and the theory predicts that it's going to be like this, and so on and so forth, Van Prassen says, ah, the theory is just correct at the observable level, and the response to that is, look, that just really seems, it seems quite remarkable that theories should consistently make predictions that are correct. that are correct, in the theory, isn't in some sense true at the trans empirical level. And, I mean, beyond arguing along those lines, it's difficult to convince somebody who's actually sympathetic to the pancrastic argument. What the explanation is, this, the existence of the, and the fact that it has zero spatial divergence, is meant to explain why you see I mean, yeah, but as soon as you get independence, that's unfair. You can't make the goal explanation. I mean, if the empiricist is going to even argue with the realist, the goal can't be explanation. Right, so there's got to be a prediction of some kind of success, even if it's historically stated. Otherwise, you can't even, there's no debate. I think there might be an impact. Yeah, yeah. This is about your, I think it's about your response 3, what you meant about some other way was some other theory, such as secondary law. Whereas what I was thinking was that some other way might be that we have... The question is, how do we know that 101 Dalmation is there as well? We see it. We have a capacity to see me. What's wrong with that as the epistemology as well? We have a capacity which has proved reliable in time. And the danger is that you're pressing for saying, I don't want to say it's some specific mathematical formalisation of the Pigeon Hall principle.

1:02:30 I agree with you that we do see it, somehow, and that's part of our mathematical experience. Right, but that capacity... I mean, anomalous can be quite happy that we have that capacity just sometimes to see things. Ah. Right. You should tell a person about ground counts and great counts, isn't it? Yeah, so I've got most 32 and great, great counts. You did just drawing on the board. I mean, there may be... The danger is that there are numbers on the board. The analogy I was thinking about during the talk, which might help you. It's a bit like some of the objections to the induction. Do we know that some children? We have a capacity, which is always to be allowed on time. For epistemology, that's good enough. Do you think that's been enough for this? I think the world has to be like that for induction. I think in order for induction. No, but epistemology. You said this was a talk about epistemology. Sure. Epistemology is what we're entitled to. What we're entitled to because we have a reliable user. Yeah. Not because we have a theory to explain that. I mean, if you get into that, you're in danger of saying that until people have a theory of colour, no one knew that they had a red book in Moscow. No, well, that's exactly what I'm saying. I'm saying that, in some sense, the student who actually did make a religion didn't know something, but didn't know that she knew it. And I'm wondering what it is that she knows. And I think what it is that she knows... Does she know the 101 Dalmatians, the 101 Dalmatians, the 101 Dalmatians or the 2? No, but I don't think it's specific. I think it's got an abstract element. I agree with you. I don't want to commit myself to she knows the axons of first order, people who are reading it, or anything like that. But I do want to commit myself to her knowing something abstract about the properties of sets, finite sets. Well, I know that's what you... that's weird... I agree. That's the second correct line question. What she knows is the specific case for hundreds, and then as soon as she sees that she'll immediately generalize. She can take a generalization, but why should they not just be afraid of that? Oh, well, we don't believe in numbers to begin with, so they can't even generalize. Because they can state. I can't say it's valid for 100, but they don't believe in 100. Yeah, but you agree, they don't have to state it. Oh, they're stated, right. Yeah, because we've got the normalistic statement of these things. The thing that's normalistic problematic is not the statement, it's the deduction. I mean, we're going to know.

1:05:00 What I'm saying is, they state it, the normalistic standard way, and then we say, oh, well, we abbreviate that, and we can just see how to abbreviate that in language of 100. How you see it is, we just have a capacity just to see. Yeah, that's kind of like a faculty. And I agree, I think there is. There is a faculty in the mathematical understand it. The question is whether that's consistent with anomalous interpretation or whether it's consistent with it. And I'm arguing that. But maybe it is consistent with anomalous Anomalous interpretation of mathematics. I'll just think about it, I'm not sure. There is some kind of general capacity there that's involved and I'm trying to analyse it that's involving an intuitive, not explicit, articulated, but an implicit understanding of finite sets and non-vetorial possibilities. Well, I'll stop now. I'll just say that the danger is that your argument does look a bit like having the same shape as saying that something he does colour vision has a graph that doesn't look colour vision. Well, I think that what, what, I should put up the German quote, because it's actually related to this, precisely this, the thing that it's highlighted here, it's a long German quote. It's the German quote from, what is Cantor's continuum problem? I don't know why I put the whole thing in. If you're curious, you want to read the whole thing. He says, the given underlying mathematics is closely related to abstract elements contained in our empirical ideas. He said, if by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things upon our sense organs, sense organs are something purely subjective, as counter-certified. They may represent an aspect of objective reality, but as opposed to the sensations, their presence in that space, is another kind of relationship between ourselves and reality. So that's really what I wanted to explore. If we were actually going down this line and talking about a capacity to understand particular instances of this kind, there seemed to be some kind of abstract element in our understanding of these pictorial representations. And Goebbels is arguing that this abstract element contains our incredible ideas are due to another kind of relationship between ourselves and reality. I don't know whether the phenomenon is still not past the view, even if what you said is true.

1:07:30 The phenomenon is still not past the view, or have an account of the proof. There'll be no alternative, that's right, yes. So, so someone would see it, and in a normal way to explain that someone would see it to some extent, but not how someone could have improved. So, maybe it could have improved the plurality of the system of building a... I just had a question about your point seventeen. Your main claim, postulating number sets and functions is practically indispensable for coming to know that certain valid inferences are valid. And this is supposed to cause a problem for the normalist, right? Right. If they're practically indispensable in coming to see this, then there's some kind of threat of invoking miracles or something to... Unless the postulated things are really there. Yes, that's right, yes. The anomalists won't postulate those things. Yeah, yeah. But it seems that there's plenty of other examples of inferences which sort of take detours through premises we know to be false, use of referring expressions to things we know not to exist, and so on, which we have no problem in conceiving as ultimately truth-preserving. example. If our problem is, well, suppose a rational Newtonian were faced with this data, what would they conclude? Now, formulating the thought processes of a rational Newtonian in terms of psychological principles might be just as horrendously complicated as the kinds of, you know, 3,000-page things that you were talking about before. And it might be incredibly much easier to sort of slip into a sort of quasi-Newtonian vocabulary We're helping ourselves to premises taken from Newtonian physics, even though we know that they're false, and this might be, like, short-circuit the inference massively, so that we can come to absolutely, thoroughly reliable conclusions about what a rational Newtonian would conclude. But there's simply not the slightest hint there that we run into a problem about,

1:10:00 what's this big mystery about how we're able to have these correct predictions on the basis of premises we know, false references to items such as mass and force on their Newtonian specifications which we know not to exert themselves. But these are approximately true. I believe the correspondence principle, so I believe that... What are approximately true, sorry, the Newtonian principles? I believe that del squared phi equals rho arises from... Well then okay, then we can just say what what would a rational alchemist conclude on the basis of all this? It's not the alchemical principles that you're taking as hypotheses, it's the fact that from those principles such and such conclusions follows. Well, I know, I know, but still the reasoning may very well proceed on these sort of assumptions of the truth of the alchemical premises, for example. Yeah, I think the only way I have to respond to this is I completely agree with you that false theories can be predictably successful. That's clear. But what I would insist upon, again, as a realist, is that I think the appropriate false theories are approximately true. And furthermore, I would say I don't really understand how to analyse the notion of approximately true. It's a huge problem exchange. But no, surely the fact that this is a reliable predictor for what the rational Newtonian will conclude on the basis of this data has no connection at all with the degree of approximation to the truth of Newtonian science itself. Oh, right, right. Because it's an abbreviation of what would otherwise be very complicated psychological and so forth. Yeah, but it's still, you haven't assumed Newtonian physics in the argument. You've said that these premises allow us to deduce these conclusions. And that's a fact that it doesn't assume anything about utopia and disney. Yeah, but I think the analogy here, well, what are the things that are actually false? Is it the beliefs of an utopia? Yeah, well, they're false, but also those portions of our reasoning which, whereby we help ourselves to the concepts and terms that the Newtonian is for ourselves using. In that kind of example, I would say that the real, well, it's not clear to me that, I mean, I think that the crucial thing is the true value of the sense in which the Newtonian

1:12:30 physics itself is actually true at once. I think that's the, and you're saying that that isn't the true value. I'm trying to trade on the idea that the Newtonian physics itself is just straightforward and follows that. Yeah, I agree. My response to that is the correspondence that I think has to hold in these cases. So in the same way, the ideal gas law comes out as an approximation for the fancy gas law in the middle of the statistical physics textbook. You can get F equals Na and G equals, you know, the gravitational force is the product of masses is divided by a distance squared from Einstein's equations. And you have the appropriate correspondence relations between the approximate laws and the laws that we believe at the moment to be correct, even though they're going to be revised. So even the laws that we have at the moment are going to be revised in the future, presumably. Maybe space-time isn't strictly a continuum or something. But nonetheless, I would want to claim that those theories are approximately correct. Sorry, I looked at my key. other people, but no, I mean, my intention for bringing forth the example was, I mean, I would, in that case, simply want to substitute a theory that we know not ought to be approximately true. Yeah, but then, in that case, if it were the alchemy example, then I think they'd just get all the wrong, the predictions would just all be wrong, presumably. No, I don't think the alchemist predictions would be wrong, but our predictions about what the alchemist will say will be wrong. That's what the problem is. We're trying to figure out what the alchemist will say. All we need to do is know about tokeness in the alchemist's head, don't we? Well, like I say, if we restrict ourselves to the processes that are going on in the alchemist's head, that would be, and that may well turn out to be, unfeasibly and completely impracticably complex in the same way... Yeah, but the truth of alchemy wouldn't have anything to do with the tokens of alchemical sentences that occur in the alchemist's head. No, but it might be just as effective an abbreviation of our recent procedures to take a detail through that as an alchemy would read. The alchemists would have an alchemical story in their belief box, and they would manipulate it in certain ways, producing sentences. All we would need is a purely syntactical analysis of the alchemical sentences in their belief box. But the choice being made is, yes, we could do it that way, but there might be unfeasible wrong, because it's much, much quicker. They won't flex or don't pretend to be often. And then immediately we'll get the answer. No, because that's exactly what you're doing in this case.

1:15:00 Do you think that every time you prove A implies B, you have to say, let's suppose we're the kind of people that believe in A? Well then, that's what, because it's the implication that's relevant here, not the... Yes, I think I agree with that. why in this case there wouldn't be any difference at all. All we have is a bunch of sentences and there's the alchemist's head, he's got some alchemy beliefs, and all we do is we inspect those and see what sentences come out. I don't understand what the problem is. I take it the point is not just about love for women. The word was used was rational. What's irrational for a person to believe? And the way you find out what is irrational for a person with certain beliefs to think is, the argument is to pretend for the moment to be someone with those beliefs. It's like John says, all we need to do is figure out what... Logically... This is how so much work. What's rational for someone to believe? How can he have beliefs in the first place? If he's A rational and B has a series of irrational beliefs, if those beliefs are in some sense rational to hold out that instance in which he holds them. We're talking about someone who's rational in terms of knowing what falls from what person. My answer is, sentence it. You don't have to, in order to prove A implies B, you don't have to believe B, A. All you have to do is carry out a logic. I wanted to talk about number 21, the star, in some sense very close to the discussion I think it's an important question how the normalist could justify a star, and if he could do it, it would be good for him. So, justify or even formulate stars, because it might be a problem for him to formulate his highest balance. But let's forget this. Let's forget this. Another point, then, would be if you think at field. Field has this conservativity problem, so you have this nominalistic theory, n, plus mathematics, and the whole theory should be

1:17:30 conservative over the nominalistic part. So, instead of z proves phi as valid, you could take, in field sense, assume that c plus n proves time, then n proves time. So it's a little bit different from star, but I think it's in the sense of what the nominalist wants to do. And field claims this is the case. And then there may well be a chance to prove this slightly modified star. Of course, you don't prove it in first-order logic, but the The means of the nominalists are not restricted to proof in first-order logic, he can add some other theories, not arithmetic, but maybe some type of individual or theory about concatenation or whatever. What do you think about this? Well, this relates, Mary's probably thinking of asking this kind of question as well. I was recently reading John Burgess and Gillian Rosen's book, Subject With No Object, again, where there's a brief discussion about metrologic, in which they suggest that field, even if this is true, in the appropriate case of the nominalistic theory and set theory, field is entitled to believe it. Just to prove it, you have to show that any model of n can be expanded to a model of n for set theory. Or you need to, if you can give a proof theoretic, But even if you can do that, then Field needs to somehow show how to prove it non-realistically. There is a paper by Field called A Non-realistic Proof of the Conservativeness of Mathematics, which I haven't read. So, I don't know what he does there. He does it modally. I'm not a friend of the modal approach, I would say. Yeah, but the modal approach, I think, takes exactly the same question. But I thought not about model proof or so, but proof in concatenation theory, token concatenation theory. Of course, this should be done. I don't know. Possible, possible. But, perhaps... It has to be possible, surely. It couldn't be the Klein-Gutman strict financiers. No, not finite. But, you see, you shouldn't do it for each formula u, because if you assume infinitely many formulas, you have to do infinitely many proofs. So, for any F-O-L formula feed, this has to be quantified, again, so the problem is to write down what a proof is nominalistically and what solidity means.

1:20:00 But as soon as you can write down formula star in a nominalistic way, there is a chance to prove it nominalistically, not logically. Yes. Yes. And then the question, how can it be that mathematical theories don't refer to anything, but still some of their consequences turn out to be true. There's no miracle left. Yes. Yes. But in order to make conservatism work well, it would have to be a nominalistic term. This is the main problem, in a nominalistic way. Absolutely. But I was also thinking about science without numbers. First of all, I think nominalism usually is not regarded to be a kind of practical issue. So it's not, they do not claim that it's a practical thing to be a nominalist, you know. It's just that they claim that they can reconstruct theories in a nominalistic way, and that something like that is always possible. It's a kind of reconstruction argument, reconstruction in principle. So, for example, if I remember correctly, even Field says in Sciences of Numbers that, um, mathematical derivations can be much shorter than purely nominalistic derivations. I've got it. Yeah. So, I think, yeah. There we are. Any inference from nominalistic premises to a nominalistic inclusion that can be made with the help of mathematics can be made. Yeah. In fact, that's why I started thinking about that. I went back and looked at that. Yeah. To recall that. Yeah, yeah, yeah. And that's not a problem for him, obviously. He doesn't care about that. It's just important to see that he can reconstruct, say, classical physics, something like that. I don't think it works, by the way, what he does, but let's say it works. He wants to show that, in principle, we can reconstruct it. And he doesn't care whether the derivations get much longer. So that's one point. The second point is, that's about this problem that you just described. but it's now in science of numbers applied to physics, you know. How does he argue for his thesis that we can reconstruct classic physics nominalistically? He argues on the basis of certain mathematical theorems. For example,

1:22:30 it's not just this result that you quoted, but it's also, you know, Tarski's results on affine geometry, various theorems in the theory of measurement. And how can hidden within the book, and it more or less goes like this, it says there are two possible cases. Nominalism is true or it's not true. So there is mathematics. And I'll simplify it, okay? If nominalism is true, okay, then the nominalism obviously is on the right side. This is that realism implies nominalism. Yeah, yeah, yeah, yeah, that's right. But let's suppose, no, that's mathematics. The mathematicians got it right. I mean, if they take it literally, And one can derive from that that we can reduce classical physics to nominalistic physics, or in the same way that we can reduce those theorems that you stated to logical truths. So if you think that mathematics is okay, then you have to admit to the nominalists that you can do it nominalistically. So I think roughly that would be the argument that he had in mind, and it might be applicable here. So it's kind of like an indirect assumption, you know? That is what is glass. Shapiro discussed this, I think, in his book. The idea is that realism implies the dispensability of man. So the whole thing can be read as a produtio and absurdum of realism. Just assume realism and then that implies. But if we go back to this thing at the end, which is in actual fact based on the dialectic, I called it, I do think that these practical issues actually are important, that the novelist actually can't. Maybe I should get the actual dialectic added. Hartree notes, that's why I got this from Hartree notes, that Hillary thinks that Hartree could, in principle, deduce it, but only in this huge amount of time. Right, so in actual fact, that's where these feasibility questions do actually come in. It's not as if, for example, Field said that you could normalize Plassand's equation by introducing these betweenness and congruence predicates

1:25:00 representing these fields and space-time betweenness. You can write down some fancy formula A using these predicates, and then this re-expresses. But it might be the case that these inferences are actually fantastically huge. They might not be in this case. But in this case I think that the feasibility issues actually do come to the fore. So, but I'm not sure is how I answer that. I do think that, I'm not sure how that dialectic goes. That kind of realism and self-imprisoned is meant to be mathematics. By the way, conservation is not true as we, Strictly speaking, mathematics isn't conservative, because if you add mathematics to sufficiently rich geometry, you could do the girdle sentence, and so on and so forth. Alright Tom. In respect to some things that Brandon was talking about in the field of Van Frassen analogy, one of the thoughts is that, of course, Van Frassen, at least in the sort of toy Van Frassen picture, he's got no problem with the existence of unobservable entities. The question is just whether we can believe in any particular ones. So, you motivated nominalism in a metaphysical and in a systemological way. Of course, if you were really motivated by the metaphysical thoughts, and these are some weird entities, and you shouldn't have anything to do with them, you've got a dis-analogy. Because, um, where Van Brassen thinks that some of these theories with unobservable can likely be true, could be true, but the metaphysical nominalists be true. That's why often you say they think it's false rather than they're agnostic about its truth. So that's going to be an important disanalogy. But moving on, one of the things that motivates the scientific antivirus is the thought that, look, they're going to be infinitely many, allegedly, infinitely many theories that are inconsistent, but are nonetheless empirically equivalent, and indeed all empirically adequate. So in some sense it's, that's based

1:27:30 to undermine the miracle thought to some extent. Certainly the inference from we've got empirically successful theory to it must be the true one, because there's going to be a whole bunch So, one question is, to what extent can we run a similar thought in the mathematical case, so you'll sort of non-standardly arithmetic, that we could have, that are inconsistent, if we think of, if we think of, if we think of, if we think of, if we think of, if we think of, Let's go. Hold it. Hold it. Thank you.