Natural Numbers / Frege & Peano / Mathematical Forms / Mathematical Definitions

0:00 The following is a dynamic in a difficult way. The Churches are the groups. Either for me to pray and we take the basic form of numerical expression to be a singular term, reparsing ordinary objective of statements of number as statements of identity, or we take the basic form of numerical expression to be a predicated concept and seek to repast the substantial uses of numerical vocabulary with which number theory allows. Now let us examine both details of the practical behaviour of the number and its relevant interpretations. First, the substandard views. The substandard form, the number of Jupiter's moon is 16. General form, the number of the concept F is N. Craigie remarks that the definite article D precedes medical terms and that those terms have positions of ordinary proper names. This statement says that the number of Jupiter's moons is identical to 16. This is the first step of working on the road of Freudian Platonism. The Neophregians here at right have articulated Freudian's remarks to give an account of numbers as self-subsistent objects. They have formulated and defended the Follick argument. Arithmetical expressions functioned in certain statements as singular terms. The statements in which they function so are true. So, there are objects to which arithmetical singular terms refer. A Meovrenian argument reduces the problem of the status of natural numbers to questions concerning the truth of arithmetical identities and the syntactical function of arithmetical expressions. One of the presuppositions of the Neoprenian claim is that the truth conditions of arithmetical identities can be fixed. The Neoprenians assert that this presupposition is fulfilled by means of whose principle, which is an identity criterion for natural dominance.

2:30 However, the crucial point in the present account is the other, the second supposition in the Australian argument, that is the availability of syntactical criteria by which we can discriminate singular terms from other types of expressions and by which arithmetic expressions can be characterized as singular terms. He and Wright have undertaken the task to present a set of syntactical tests by which singular terms can be recognized among several kinds of expressions. However, such an attempt has been broadly criticized on the grounds that the alleged test do not succeed in excluding plenty of troublesome and misleading expressions which only apparently behave as singular terms such as a constant or obvious. Let's see the complete and sharp distinction between singular terms and other expressions is a very ambitious task which has not yet been met. So the demand of a set of appropriate criteria which set necessary and sufficient conditions for a term to be a singular term yields a special difficulty for the substandable account. account. Now let us turn to the adjectival account of its interpretations. Here is the adjectival form. There are 16 moons of Jupiter. The adjectival use of argumental expressions might support the claim that numbers are properties. So, they might be properties of physical The main difficulty of this approach has been stressed by Frege who argued that there is an obvious difference between numbers and properties. For example, a common property characterizes a physical mass in a definite way, whereas a number does not. The way a number characterizes a physical mass depends on the special manner we choose to consider it. A crucial point to stress here is that numbers should not be considered independently of a concept f by which we choose to view a physical mass. For example, if we speak of the concept pack of game cards, we apply number 1, but if we speak of the concept game cards, then we have to apply number 52.

5:00 So, we could have an interpretation according to which numbers would be properties of physical collections, not properties of physical collections, but properties of concepts. Numbers tell us how many instances of the concept there are. In that case, arithmetical expressions might be construed as one of the forms of concepts. However, this interpretation faces a serious difficulty too. Although identity is well defined in case of arithmetical one to bias, the interpretation of another such properties appears to be blocked by an ultimate metaphysical problem. If numbers are construed as properties then we will have to deal with property identities since numbers systematically occur in mathematical identities. So, if we interpret numbers as properties then we will have to be involved in the traditional metaphysical problem concerning determination of identities between properties. The difficulty is that identities between properties cannot be determined among biggest people. can be construed either intentionally or extensionally. It is possible for two properties to have identical extensions with different intentions. To construe properties as extensions, you have to ignore their intentions. Some philosophers have dealt with this problem by treating arithmetical identities in an extensionary way. For example, Penelope Marty treats property identities in terms of a scientific law like coextensiveness. An extensional reading over medical identities or property identities leads us, however, to the first interpretation of novels as objects, since extensions are usually themselves taken to be objects. So the ultimate physical problem, the determination of identities between properties, yields We have so far presented two forms of syntactical behaviour of arithmetical expressions and two types of interpretations or natural numbers as well as the basic difficulties those interpretations

7:30 face. Which of the two accounts describes numbers? A reductive approach perhaps might help, so we have the following options. The first option of the redaction strategy, the substantival form can be reduced to the adjectival form. In this case, we have to assert that the numbers are eliminated. Take for example the sentence, the number of cats in our house is two. This can be paraphrased as, there are two cats in our house. The latter sentence can be written as there is an x and there is a y such that x is an f and y is an f and x is different from y and from every zent such that zent is an f then zent is x or zent is y. F here is the concept cut in our house. This example shows that quantifiers consist a useful instrument for eliminators. One of the main tasks of mathematical languages has been the formulation of scientific language in a way such that mathematical terms are eliminated. So, if they are eliminated, then they are not genuinely singular terms. So the imperialists assert that inclusion of the claims have not a biological convenience to mathematical entities. There are only concrete facts in which our statement becomes true. The second option. The second option of the redaction strategy consists in that the adjectival court can be reduced to the substantive court. So, this option is odd enough since it expresses a kind of agorian view, so long as it is supposed to reduce statements without ontological commitments to abstract entities to statements with ontological commitments to such entities. Some philosophers think that such an option must be blocked by a demand of ontological economy. Allen Mass Greig also claims that numerical quantification is a perfectly well understood notion on its own. to reduce it to any substantive account from it to abstract. Yet, a mathematical realist should not at all be annoyed by a second option. She should adopt it as an inflationary way of reading the alleged paraphrase. However, to take the substantive form as a deduction

10:00 basis, this apposes that realistic position has sufficient grounds. Nevertheless, semantic Some paraphrases cannot be conclusive in either direction. Some could read them in an inflationary way and others in an inflationary way. We will try now to go for the option that the two forms of syntactical behavior, writing and concepts, are equivalent to each other. So the third option will provide a kind of double vision of the situation. An alternative option, the substantial form is equivalent to the actual form. This option can be actually based on the principle NQ. A proof of this equivalence is presented in the Reasons Proper Study by Principal Right NQ shows that for each Prakian number N, it is established that N is the number of the concept F, even if there are exactly N, S. Independently of the very interesting reasons the Neoprakians appear by principle NQ, my concern with reading this paper has to do with the fact that according to the R.S. principle, the substantive undercreditative views about medical expressions appear to be equivalent to each other. Medical expressions appear to be both singular terms and predicative concepts. So our natural numbers after particulars or universals, this situation reminds us of Frank Ramsey's claim that there is no real distinction between particulars and universals to be asserted on syntactical grounds. According to a Ramsey type solution, which of the two forms, the substantival or the adjectival, we will choose is a matter of literal style or of the point of view from which we approach the mathematical fact. Of course, a dissolution would deflate our dialogue. A more metaphysical audit response, however, could be. Perhaps, syntax is a pure

12:30 guide to ontology, so we have to seek for other resources to investigate the ontological status of natural numbers. Numbers still can be entities which possess two different modes of presentation in language, both the objects end as universals. Moreover, those modes are equivalent to each other. So, we needed a physical account of what an object is. According to that account, an object is an item which satisfies certain identity conditions. But then, numbers are undoubtedly objects since they satisfy a certain identity criteria, of his principle. However, the fact that natural numbers are objects does not entail that they are particulars. The possibility cannot be excluded that they are universals too. There is at least one philosopher, Jonathan Law, who allows for entities which are both objects and universes. So our metaphysically loaded response might be to keep on the equilibrium function between the two forms of mathematical expressions and observe that those forms are indicative of a double metaphysical status of natural numbers. Numbers are objects, namely the objects of reference of expressions so behaving like names, and they are universals as they are scripta of expressions so behaving as predicates. The fact that they are objects, namely items with satisfying identity criteria, does not exclude the possibility that they may be universals as well. Then the two forms of their syntactical behavior are indicated on the term space of their metaphysical Thank you. What I was saying is that among the two forms of syntactical behavior no prevalent to choose. And among the two interpretations of natural numbers as objects

15:00 and as universals, there is no prevalent to choose two because they both face difficulties. So this principle, NQ, is indicated perhaps on the fact that natural numbers have a double metaphysical status and language is suggestive of this double status at the metaphysical level and so I think that we need not a syntactical but a metaphysical answer to the question of what an object is. So an object is an item which provides for its instance and instances is evaluate criteria of identity. Natural numbers satisfy a criteria of identity, huge principles. So I think that they are objects. But this option does not exclude the possibility that they may be universal to. There are other physicians like Jonathan Lowe who asserts this possibility. And I think that I have to think more about this. Do you think it would be also, sorry, this is only about numbers. I think that the same might be Carlos. I'm not sure, but I think that the colors' names behave in three equivalent forms in language. We say that this table is green, and we say that green is beautiful colors, so green is predicate for a similar term. But within mathematics, it's certainly the numbers are just one case, but there are zillions of cases where you have things that are objects and properties at the same time. Jonathan Lowe says that there are objects which are universal and these are natural kinds. But I am thinking of the idea that natural numbers are objects as the reference of expressions

17:30 of the claims in real terms and they are the astrita of appropriate credit gates. So they are universal as properties of concepts not kinds. of concepts, or sort of concepts. I may start a question, because you were saying the numbers are properties of concepts, and then that suggests to me that you want to distinguish between properties and concepts. Does that make a difference, or would you just say the property of properties? I take it in the Freudian sense, we have a sort of concept, for example, cats in our house, cats in our house, this is the concept, and the number two says that this concept has two instances, so it characterizes the concept. or the property of being a cat in my house. But we should also say that there are properties of the second level. Yeah. The second level properties. I was wondering whether there's a special reason that we were talking about concepts instead of properties. Instead of properties. I think it's the same, but I'm not sure that Frege identifies when it comes to doing that appropriately. And maybe you should play this in a minute. I'm not quite awake yet. So, the last slide was the idea. I want to continue working. I have to think about how it is possible that numbers are both universals, properties of concepts and objects, I'm sure about objects, but I have to think more about properties because there are many interpretations of natural universals properties. Properties of physical collections, properties of sets, properties of concepts, and collective properties, and other recent interpretations.

20:00 This interpretation places the difficulty, I said it before, if we construe national other size properties, then we have to be involved in this traditional problem about how we can determine identities between properties, because we can't construe a representation But if we choose an extensional reading, then we will return back to the object interpretation. So this is a difficulty for me. But do you know already what objects are? Objects are realities because there is a spike in the object criteria. What's the identity criteria? A huge principle. For the numbers. About the numbers, the number of the concept is identical, it provides great conditions about when two items are identical to each other, in general, the identity conditions. Please, I don't think we can pronounce your number six years, please. I don't know how to phrase this. Even within mathematics, you have problems with trying to resolve the questions of whether numbers are properties, even within numbers here. For instance, if you take something like a prime number term, how many primes are there below 10? So your objects are numbers. And then the number of primes less than is a property. And you're counting that. SPEAKER 2 Yeah. So the same sentence you have, you're talking about the properties are numbers too. They're all the same thing. Whose principal provides such situations? What are the numbers? That's fine. But in the theory itself sometimes you'll have properties will be the number of...

22:30 Yeah, we can take the property, prime number, and then say that the number of the prime numbers is an object. Which characterizes, it's also a property or the property. It characterizes the property from another. Sure. I was just saying, you were mentioning if we take numbers as properties then we have a problem as to how we're going to... No, it's a double status. They are objects and properties at the same time. I agree, I agree. Thank you very much. This is depending on the context, in a way. That's all. All right, all right. Not Jane's face to mold the facetive. So Colin says they can be whatever they have to. in order to participate? Nope, you're on. Just start. So, just use this, okay. Yeah, okay. Okay, so I think you were presenting this morning,

25:00 from the University of Vienna, and almost the same topic? The topic of my talk will be Frege's conception of quantifiers and more specifically his understanding of the notion of scope that comes along with it. There has recently been a fair amount of discussion on general issues concerning Frege's conception of logic and logic expressions and the intended semantics of it. I'd like to focus on one specific issue, and that is the question whether or not nested quantification, that is multiple chlorophyte, can be expressed within Tregers' logic that is somehow similar to our current understanding of nested quantification. There is an influential work by Michael Domet. He stated that Reagan was the first to introduce the tools for use in multiple generality. Against this, in a number of papers by Yachim Hüttemberg, he suggested that due to the specific semantic conceptional quantifiers as higher order concepts or truth functions, this is not possible, unless the quantification is not possible. So I'd like to take up this debate and focus on the syntactics, the notational specifics in Frege to see if there are any problems about expressing multiple generality. And I'm trying to do this by taking a kind of detour and looking at a correspondence between Frege and Piano from the early 1890s and see what is discussed there on this topic. So the structure of the talk will be as follows. I'll start with a brief exposition of Frege's definition of quantifiers in his Begriffschliff, the concept script of 1879, and I move on to this correspondence between Vega and Piano, where we make some remarks about Vega's formation rule for his foreign language in the Grundgesetz

27:30 that I have made it, the basic law of the arithmetic, and then finally I'll say something on what Piano has done, Piano has proposed on the topic of nested quantifiers. So, as it's generally known, the Vega first introduced systematically symbols for expressing generality in English with Liebschrift. I'll just bring you the quote from section 11, from the first part of it. It says that what's interesting here is that he introduces the symbol for generality not by one symbol, but with a combination of two sides. So one is this concavity on the horizontal line, and the other one is a specific type of variables, the German monocropic variables, which express the generality. So the quote goes, for all A, V of, A signifies the judgment, and the function is affect whatever we take as its argument. And existential judgments are then expressed by the generality and the two negation on each side of the concavity. Now what's interesting about this definition is that Frege here gives a kind of contextual definition of a quantifier in terms of the proof conditions of the sentence involving it. So this becomes clear once you compare to it the definition for quantifiers he gives in the Wundgesetze 15 years later, 14 years later. Here the central passage is this, that for all A, if A, you know the true, if the value of the function phi of xi is the true for every argument and otherwise the false. Now, in contrast to the first definition, what is done here is that the quantifiers is, or the quantifier is defined by the truth conditions of the sentence in which it shows up.

30:00 And he's using here, in contrast to the Begriefschrift affinition, a semantical phrase, the connotation, and the truth value, the true or the false, which are seen as objects in Frege's system. So, irrespective of this semantic side to it, there is a particular and interesting syntactic specific about his definition that is somehow left unspecified in Frege. And this concerns the specific function, the concavity and the gothic or term that have in his notation. So the question is, is there a kind of working division between the two, whereas the concavity stands for, is the sign for expressing something, or the variable is the sign for expressing something. There is this, next to the there is an asset in which this, the functions seem to be clearly delimited. So, and this is where he introduces the notion of the scope, what he calls the complete of the judgment. The concavity delimits the scope that the generality indicated by the letter covers. The term that it retains fixed meaning only within its own scope, within what one judgment, the same letter can occur in different scopes, without the meaning attributed to it in one scope extending to any other. So it seems clear from this that what Frege intends here is that the concateness used as a sign for identifying the scope of a generality, whereas the specific variable, the German variable, stands for the generality to quantify itself. However, in a recent paper, Volker Peckhaus has proposed, in the reading of this, proposed an invitation of Regis Bigliefschliff, that it's the opposite to this. So he's saying, basically, that the concavity itself stands for quantified in the modern sense, in the current sense, whereas the German letter indicates the scope of the quantifiable.

32:30 And he draws to, there are some actual evidences in late writings, in the late 1980s, which seem to support or give some support to this reading. However, to the beginning, this doesn't work, and there is one simple but conclusive argument against it, and this is the introduction of judgments with multiple generality. So here's the interesting quote. The scope of a German letter can include that of another, as is shown by this example. And this basically for all E. E of AE implies A of E. So this is the syntactical sign for the implication relation. In that case, and it goes on, in that case they must be chosen as different. We could not put an A for an E. So what's important here is that this last condition has no that has no implications on the scope of the formula. The scope between these two generalities is sufficiently specified by the graphical position of the concatity in this formula or in this judgment. So in this case, the e, the quantified binding g is in the scope of the quantifier binding A, and in the alternative case where the concaptive would be here after the implication sign, it would be that the A would not be bound by quantifier. This is interesting for two reasons. One, it shows that Frege is actually introducing the notion of, or the idea of the master quantification here. And the question

35:00 is whether it can, the question is whether it is comparable to our modern understanding of it. And to see this, I would like to contrast Frege's theory with another one by Keanu, And what Keanu did was independently of Regen, without knowledge of the degree, define or introduce an alternative way to speak of use, speak of generality in judgements or in formulas. And he first does this in his famous paper, Arithmetica's Principia of 89, in which, as you know, the axiomatic structure of arithmetic is first introduced. And here in the first part, he also introduces the, in combination with the sign of the implication, introduces a method for indicating generality. So I'll just read it as the passage goes to this. The sign Horshue means one deduces. If propositions A and B contain the indeterminate objects X, Y, et cetera, that is our conditions between these objects, then A Horshue subscripts x, y, etc. B means whatever x, y, etc. may be from the proposition A one deduces B. Now, this is obviously a completely different symbolism using a speaking of generality than Freges. And what interested me was to see how Freden piano themselves, how they use the difference, the notational differences. And an important, important, interesting source to see how they use the different notations is an undiscussed correspondence between the two from 1894 to 1896, basically. And so I cannot go into the details of their discussions

37:30 on the quantifiers and the definition of the definition, but I would just like to focus on one interesting critical remark thinking with regard to piano symbolism. And so what he's saying is that he acknowledges that the improvement in his point is the improvement of Hege's notation with regard to the notation in Imbul or Schroeder, but then he's saying your expression for generality might show to be less generally applicable than mine. And I do not know whether the scope of generality, what he calls the Gebiete argument, will be clearly delimitable in all cases. So, Frege leaves unspecified what he means here by the term in all cases, but given his own introduction It seems more than a good guess to say that what he has in mind here is the biggest lack of, what he has in mind here is formulas or judgments or propositions involving more than one generality. that is expressing multi-generality and the different kind of relative scopes that come along with it. So what Vega seems to be saying is that due to the lack of formation rules and clear rules concerning the use of pontifiers, Piano's notation does not allow to express these more difficult more advanced type of cases of propositions involving multiple generality. Now this is interesting for two reasons historically. One is one can see from this and from other passages which I cannot quote here what importance Pflege attributes to the fact that a formal language like his

40:00 is able to able to capture multiple generalities. That's the complication. What is interesting is to see whether his critique or his stuff can be justified. It's true. So before coming back to Piano's own symbolism, let me just make some remarks on Frege's evolving views on his language that seemed to support my first assumption. And it's interesting that at the time when he formulated his critique of piano symbolism, he has just published his Incaset, the Arigmatic Basic Laws, and here he gives a really detailed articulation, specification of the formation rules, syntactical formation rules for his language. So the central section is the section 13, the first part of the book, and what he does is actually define two methods of forming names. So, yeah, what has to be said is that Faye's basic vocabulary and formulation rules differ strongly from our current definition of vocabulary, or logical vocabulary. So this basic signs are names, that is proper names and functional names on one side, and from these one can build compound names. So the first direct methods of forming names includes four rules, or four possible rules. One can form a compound name by inserting a proper name T in the first level one place functional name. One can do the same with the second one can form a proper name. This one by inserting a first level one place functional name in second place a second level one place functional name same for third level

42:30 concepts, and then one can formulate a first-level one-place functional name by inserting T in a two-place functional name. Now, with these four rules, these four rules are are severely limited because most parts of first order and high order polyadic logic are not expressable in here. So what he does is he introduces a second method of forming what he calls complex names. And this is done by a kind of variableization. So what he does is, what he says, what he can do is take out a simple name T from a compound name formed along the rules 1 to 4. And by this I'll just show you a simple example how this is done. So for all x, for all y, f of x and y, you can add the f of a and y by using from f of of x and y using rule 4, then you can insert this in the second order concept, font 5 concept. And then you can take up the e using rule 5 and get a free variable again. And then you can introduce this part from this construct, let this one, this part fall all in the second-level concept. So it seems as if they actually introduced the second indirect method of forming names to enable formulas like this and much more complicated. So what about Piano himself? The most systematic treatment of this is found in a little French book, La Notation de Logique Mathematique. And here in section 18, he gives kind of elucidatory remarks, how he intends to use multiple generalities within one proposition.

45:00 And it's interesting that Brege does not mention this part of the book. So, I think my time is nearly up, so I'll just show you what he does. So, he quantifies this first formula and has two quantifying expressions here and another one here. So this can be paraphrased by this. this. And then what he does in the second step is a reformulation of this. He's taking this implication of the sign out of the formula and is introducing a new sign. And this is what he calls the absurd, which is closely or similar to Frege's The False. And by using this and the identity sign with the subscript of Y indicating the generality plus the negation sign. So with the combination of those three, he's able to articulate something that we are now symbolized by an existential quantifier. So he's able to do a formula including negative quantification with mixed quantifiers which is not found in Frege's Begriffschrift. He's just showing how to use this. For example, in 6, he's showing that you can, this means there exists an x, and this means there exists an y, and he shows that you can change the quantifiers here, the order of the quantifiers here. So, what I was trying to show is that given this symbolism, these different types of formulas, including nested quantification, can be symbolized in Piano's notation. So what I tried to show was basically three things.

47:30 One is that both Brege and Piano put a great emphasis on formal language capacity to formulate or codify multiple generalities. Secondly, Brege's critique or objection to Piano's notation does not seem justified, given the examples I've shown. And thirdly, that Indo-Piano's notation might seem uneconomical compared to a current standard. It allows a much more flexible use of formulas, expressing formulas like this than Paget's notation. So thank you. I'm not familiar with Piano's notation, and it was now very quick for me, but how about such formulas as, for example, for all x, brackets, for all y, if for all y, f, x, y, then for all z, g, x, x. So, I mean, if you have within the universal quantification an implication where two other universal quantifications occur, but each only limited to one side of the implication. That seems to me, from the first view, difficult in Piano's notation, but maybe I didn't see how ingenuity. I have to say I don't know if it works. The thing is with these two notations, Freges as well as Pianos, you have to just try it out and construct, develop formulas and see if they work. So in this specific case, I have no answer to this at the moment.

50:00 I would have to try it out. But the thing about beta symbolism is that here, too, once you move to more complex formulas, you start to have real difficulty, get into real difficulty in expressing, in using multiple you're starting to get into real difficulty, kind of giving a graphical presentation of it. And also with the formation rules of Grundgesetze, which I tried to just give a brief a brief exposition of it, once you start to construct foreign names which have to be formed using a combination of the direct method and the indirect method, things get much more difficult. This is, for example, by a point Richard Tech has pointed out in a recent article set set. So in Plague 2, you have difficulties once you get more complex examples. Just a short historical question then. So who clarified the use of Pythars as you know it now? Is it Russell or is it Robert? Yeah, this is really a multi-dimensional question. What has been pointed out and what is true, I think, is that first with, so, and I have to say this, this is part of a larger paper I'm pretending to, I'm working on at the moment. What I haven't mentioned is, and what is probably even more interesting is the semantic side to this in Frege's logic. So one could argue that Frege gave the syntactic sufficient to introduce quantifiers and also nested quantification. But it's due to the semantics, especially the semantics of the quantifiers as high order concepts or truth functions,

52:30 and as well as the compositional principle. So in his, let me just show you this. So this is the section in Grundgesetz where he introduces formation rules. But at the same time, this section includes a recursive semantics for compound names. And what he has here is a compositional principle, a strict compositional principle working here. And one would argue that it's due to this compositionality that an understanding of nested quantifications, it now is not possible. So to come back to your question, I would say that it's a very good guess to say that Hilden or Scholem, who understood the existential quantifiers as something like a Scholem function, really gave a big boost to our current understanding Thank you very much. And our morning session is Jean-Pierre Marquis from the University of Montreal. And he will speak about mathematical forms and forms of mathematics on monoppy types. Okay, thank you. This is going to be a glimpse of the subject. It's a huge subject in highly technical. You know, my goal, my main goal is simply to whet your appetite and convince you that there was something interesting going on in this field, philosophically interested. So let me first start with the background. The basic fact is that mathematics in the previous history has been developed in a framework

55:00 which is set-based structures that are classified up to isomorphism. This is, I think, taken as a starting point. it is well-known. Assume that in that context, an isomorphism is a bijective function that preserves the appropriate structure. This is what I would call the purely extensional point of view. There is an ambiguous aspect to what I'm saying here. I might clarify that in a minute or later, if you will. In this context, given an entity with property P, then any other isomorphic to it, where the layer is based on an underlying ejection as also a property P. So this is just a classifying of the isomorphism in loop theory or topology or another field. So here's the ambiguity. Here we're working up to isomorphism. There's also another sense in which it is extensional, it is a more logical sense. So given a set and a property p, so this is the interpretation of the predicate, the elements of x having p is a subset of x, and therefore there, the identity cartoon is the axiom of extensionality. So it's not called to isomorphism, it is a set of objects that is the interpretation of the predicate. So this is also extensional, but in a slightly different sense in the sense I introduced earlier. But in the working day, so if you're not in a logical interpretation, but just in the mathematical realm, you're working up to isomorphism. So, in this case, the latter case, the criterion of identity is extensible. So, the extensible point of view has become what I call a form of mathematics, and it analyzes mainstream philosophy of math and has occupied the forefront of the field, and I think this is also uncontroversial, or if you want to argue, we can afterwards. There is an alternative in philosophy of math and also in math, but all in math is really just in terms of intuitionism and its constructive variance. And they are often seen as being based on a devian form of the mathematics whose philosophical basis is considered to be dubious or with the description of mathematical objects at odds with what most mathematicians would take there's something missing here as standard. So the idea here is that, yes, okay, so if you take the set-based mathematics, everything is done up to isomorphism in an extensional

57:30 point of view. You have a classification based on that. There are alternatives. The alternative is based on constructionism or intuitionism. Philosophically, we have discussions about the basis of this, or, well, it's at odds with most of, or maybe some results in analysis, for instance, are not, well, have to be translated into the constructed version, and so on and so forth. Now, what I want to claim here this morning is that within mainstream mathematics, there is a form of mathematics that has developed slowly but steadily from approximately in 1950, a little earlier, I won't say anything about the history of the subject, although it's a fascinating subject in itself, which is becoming an important research field with applications in numerous important domains, for instance, in algebra, in physics, also now in unified field theory, you're using homotopical methods, and more and more, and mathematics in general. It's also more important in algebra and in some cases. It is extremely general. That will be able to show you who could serve as a foundation for mathematics. And it is based on a non-extensional point here. Who could serve as a foundation for mathematics? This is a project. There's at least one mathematician who has actually worked out the first sketch of how it should be done. This is Robotsky from Princeton. He has in 2002, he has given talks recently on a formal theory based on homotopy theory that could be used as a foundation for mathematics. So there is something that could actually develop. The interesting thing is that this alternative emerges not from an a priori philosophical conception of mathematical knowledge but from a basic and fundamental notion of geometric all of the types. However, and certainly unexpectedly, it turns out to have connections with constructivism and non-expetitive approach to mathematical concepts, and that I will briefly indicate further at the end. There was also recent work done on the logical aspect that reveals this explicit. Alright.

1:00:00 So, this is the background. I've been playing. Now, I want to tell you a bit about what homotopy types are. And this is going to be very sketchy. I hope just gives you the feeling. Here's the definition first of a homotopy. So we have to start from that. So you start with spaces. So you just have to have an informal notion about a space. two functions between the spaces that are continuous. Now, a homocopy between two functions f and g has to be a continuous map. It's defined from x to y and with a parameter. You can think of this as time, zero, start, time, one, the ending time. And when it does, h with z does, it starts at f and it ends at g in a continuous way. So, you think of this So this is usually, so if you're working the real plane, a very simple example would be something like a function like this, and then you'd take a constant function, this would be f of x, this would be g of x, and then h would just take this, and at 0 you would have f, and at 1 you would end at g of x. So if you have a picture, actually, a movie, that would start, an animated movie, that would start with F here, and then you would just slowly see this coming down, flattening this thing, and it would slowly go to G with N. This is a homotopy. It turns out that all functions from R to R are homotopy, or are homotopic. In this way, there's a homotopy. So, as I just sketched here, a homotopy is a continuous deformation of F into G, and we say that whenever there is a homotopy between F and G, the maps are homotopic. An abstract representation of what is going on, so this is the representation of the real plane, but it's much more general than that, is something like that. So you have a space you have two maps F and G going in the same direction, and the homotopy is a map from F to G. So it starts from F and it ends in G. So this is a very simple picture that you have to bear in mind. But the important point, and this is a very crucial point here,

1:02:30 is that there are many different homotopies between two maps. Now this is something simple, but it has important consequences on what we're going to see later, so keep that. Now, homotopy types, so what are those? So you take two spaces, and they are homotopy equivalent, and there are continuous mass going in the opposite direction now, from x to y and g from y to x, such that the constants are homomorphic to the identities. They are not equal, so that would be the isomorphism. They are homomorphic. So if you take an abstract representation of this, it gives something like this. So you have your spaces, you have f, g, you have the identity map on each spaces. Then when you compose both maps, you have that, but you can actually deform it continuously into the identity. And on both sides. Whenever you have that, we say that x and y are of the same homotopy type. Now I didn't say this, but being homotopic is an equivalent switch. So it's a sort of equality. Being of the same homotopy type is an equivalent switch. So it yields a form of equality also. Okay? In the standard topological context, you would have the notion of polyamorphism between spaces. And this is a bijective continuous function such that when you compose, they are equal to the activity. Now, homotopy equivalent spaces are related to one another in a way that there need not be bijection. So it is an equipment solution that is not based on the bijection. And it is radical, right? So again, some important things I want to underline immediately. There are in general many different homotopies between two different apps. Thus, the identity between two entities is not really an all-or-nothing affair. So if you have two spaces of the same polytopic types, there are various homotopies existing between them. And a lot of information is involved in the identity itself. Okay? So there is more there than

1:05:00 just it's equal or unequal. There's a whole lot more. So to show you how radical the notion is are radically different from homeomorphic spaces. I'll give you a very simple example that illustrates the fact that two spaces can be of the same polynomial type without being bijective to one another. So if you think of one point space, one and the real line are of the same polynomial type. There's no, there's no dijection between one and the real line, but they have the same homotopy type. From the homotopy point of view, it's the same basic thing. A polyhedron is a space I'm going to work into a simplex. I just want to illustrate this with another example. So this is a point that's a zero simplex. A line is a one simplex. A triangle is a two simplex. And then you have a three dimensional solid is a three simplex. So these are simple spaces that you construct. When you look at them, from the homeomorphic point of view, if you look at bijections, you get your point of view. If you look at them from the from a homatomical point of view, they correspond to basically numbers. So if you look at homotomy types of finite, polyhedron of different dimensions, you get numbers. Basically, natural numbers are recreated in some sense from this perspective. So this illustrates beautifully how you can get various entities from that point of view. So the claim here is that homotomy types are abstract forms of space, and more generally abstract mathematical forms. I'll say a thing They are fundamental. So I can't argue, I'll just quote a few things about this. Here's Baos. How do you pronounce his name? How would you pronounce his name? Baos. Baos. Thank you. So all the types of polyhedra are archetypes underlying most geometric structures. Here's an illustration of this taken from one of his papers. I won't explain this. These are all kinds of spaces related to one another by founders or forgetful founders between them. But at the very bottom, we have bio polyhedra with their homocopy equivalent. So at the basis of the whole constructions, which is in each case you have mathematical theorems related to them,

1:07:30 you have homotopic equivalents of polyhedron that are up here. So they are fundamental in that sense, too. They are fundamental in another sense, also. I mean, all this implies the fundamental importance of homotopic types of polyhedron. There's no good nutrition in the vertices. There's no good nutrition in what they actually are. This is from here. But they appear to be getting these as genuine and basic as numbers or nots. The question here is, where are two homotopy types identical and what is the criteria of identity for homotopy types, and this turns out to be a hell of a problem, because it is very hard to tell when two homotopy types are different, and in fact the whole theory of homotopy theory is just a search for, a way to tell when two homotopy types are different. And basically, you have to look at what they're called invariants of homotopy types, and there are, I define it, the only numerical invariants that there are are the dimensions, and their degree of connectedness, I'll tell you that they are, but they're very intuitive here of what's going on. But then, to get more invariants, you have first to define what are called homotopy groups. And these, how do you find it? This is very geometric. The And then a lot of the group of the space is this, yeah, this is up to homotomy. The homotomy is from the N sphere into the space itself. So you look at pictures of, if you're in the first homotomy group, you have circles that you map in the space up to homotomy. And that will, you can construct a structure, a group of that, and gives you an invariant of the space. And these now reveal the properties of the homotomy types. And if two states have the same, well, it's more subtle than this, they're not enough to classify the homotopy types, but you have to have those at first, and then you add more structure to get the whole thing. So, the homotopy groups themselves are not enough to get a classification, even the homotopy types are fine, you know. And the last indication that they were fundamental, and this turned out, it was discovered later, homology and cohomology groups, which are also invariants of topological spaces, they can be defined from the cohomalty groups, and this is another indication that cohomalty types are basic. The whole enterprise of logical and topology, okay, this is what is underlying

1:10:00 the whole thing. In fact, we now know that the whole thing, oops, I'm going to turn back here. Sorry about that. The whole thing is, is, can be lifted to the abstract level which was done from the late 50s and 60s and even now. Homotopy theory, what I've described to you till now, was essentially for typological spaces. Well, you can grow abstract and define types for algebraic and combinatorial entities, roughly, and you can characterize them axiomatically, right? And therefore, you get to a purely abstract notion of homotopy types, and this is why I was saying earlier that they describe abstract forms of bath-like logics, you know, only abstract spaces, abstract forms of spaces. Now, let's come to logic and non-intentionality So a recent work by, very recent, it's on the Math Archive, their paper is available there, Michael Warren and Steve Audi makes a direct connection between homotopy theory and logic, and what they show essentially is this. There's a connection with Martin Loew's intentional time theory, and the basic claim of their paper is that this is, a whole category is an abstract setup to do homotopy theory and define homotopy types in the abstract form. are the logical spaces formed in the model category when you look at the program. Any model category is a model in the logical sense of that expression of a four-confrontal Lewis intentional type tree. Now, the interesting thing here is that the key to the whole thing is what is called the identity type, between terms A and B of type A, right? And it turns out that this thing is basically the object of all paths from the unit interval to the type A. All right? Between A and B. So it's just a whole time between A and B. And the key thing is that you have here is not identity A is equal to B or it's not. A is equal to B in various ways. And so you get out of the purely extensional point of view from the fact that there are many different ways in which A can be equal to B. And this is included

1:12:30 in the type itself. So similarly, given the type X in the property P, there is more than one way to null check A can have the property P. And this is again because there are homologues in the proper set up between A and the other numbers. And it's more than just M1. Okay, So this is how you get into intentional type theory and how you actually get around, so to speak, or you step out of a purely extensional framework in the logical sense. It was clear in the mathematical sense earlier. So in both cases, you are. So to conclude, both of the types are fundamental and abstract, and I've stated my word for it, but this is certainly clear now in at least algebraic topology, but it's becoming more clear in general, because you can define the natural numbers, the integers, and all the other discrete collections or entities from them. The interpretation of identity, properties, and relations in this context contains more information than in a purely extensional context, and you can work with it to get differences that are unavailable in a purely extension context and the last claim is that it yield a genuinely new form of mathematical knowledge because the mathematical objects that you're working with are not classified up to isomorphism in the extension sense of that word Yes, please. Yeah, a very general question, Jean-Pierre. This issue of the distinction between the intentional and extensional treatment of identity as it connects with the question of classification up to isomorphism. I understand from what little I know about this work that this is actually connected in quite a deep way with the behavior of the diagonalization. Yeah. Can you say a little bit about that? Well, I can't say more about this, but what you just said is that the identity, is that what you had in mind? Yeah, exactly. What I had in mind was the difference between Law of Veer and Leibniz's identity, but this would take us into two technical areas. So the identity is focused on you with this.

1:15:00 So the fact that here, what you have, you have to construct a picture, and this is an hybrid object. We're getting to take it from you. Yeah, it's very difficult to go into this area without getting to take the details. I think if Audi and Warren Spraymark, what they do is to interpret the types of vibrations, to make sure that the identity type is a vibration, and then the path object actually does factorize the type. That's underlying, if one gets a philosophical payoff, the deeply geometric nature of the source of these logical notions. yeah precisely but I just wanted to underline that point for it philosophy This question, it's entirely unfamiliar to the . I don't know. Maybe, but I didn't quite get how hard that kind of theory could be on It could be an enterprise, but somehow you couldn't get the idea. Well, I didn't show you what happened. That's clear. I would be, well, can you give me your idea? Well, the idea is that you start with a type theory. So you wouldn't start with first orders logic. Then you start with a type theory. Your types would be homopathy types, but then what you get that you can actually add hyper-national numbers, hyper integers, and you would have axioms for , and then you could actually get a sort of equivalent theory which wouldn't look like a setup, but in which you could define all the mathematical basic, natural real numbers, and then you can proceed, not in the usual way, to change a lot of stuff, that you could actually do a lot of mathematics in that context, in the way you do it. And there is now a formal, Wawatsky presented a formal lambda calculus in which he showed how you would define the basic mathematical structures with this framework. So I think you get different things. You are able to fall back on your feet so you can get back to the national numbers and stuff like that.

1:17:30 So that's how the detail that you work out. But there's this piece of sketch that I would put in there. Any references? Actually, there are very few references. I don't know if it is. Perhaps. I mean, Jarlutsky's work is not even written. It's a series of . It gives you notes from that. On the other hand, it's also related to a whole lot of research done on algebra. So it's a lot of good guys. It can be treated algebraically, and it feels, how do I call it, higher order, or higher initial category. This is also . Any other questions? OK, thanks. Thank you.