Mathematical Knowledge (contd.)

0:00 And so, on a first approximation, my approach could be simply described as, by name, a person of a terror, where we would have something similar to what the teacher had in mind. a quintuple, or maybe only four of the elements in the quintuple, and we name it a framework. And let me now consider a pair of two things, an agent and a framework. In order to make things simple, in a more careful presentation I will be making more positions and discussing myself in more common front picture, but today it's enough to simply consider this pair of elements, an agent and a framework, it's very resemblance of the picture's idea. So I would take the practice to be consistent in both, actually. Now for the agent, there are many possibilities, but I would like to mention at least two. So, the agent can be analyzed at different levels. One of them is simply to consider normal agents, so endowed with typical cognitive abilities of the kind we take for granted in developing mathematics. And actually, for everything that I'm doing research on, this is enough to consider simply a normal agent in that sense. We could come much closer to actual historical events by considering concrete agents. In that sense, we could consider, for instance, some kind of simplified model, of course, of historical agents with their specific meta-mathematical views and their research agendas, for instance. So, as you may see, I will take the better mathematical component from Kitscher's winter ball and put it on the side of the agent. But anyway, I will not be talking about that. But still, we also could consider the developments of this kind of model in which we consider communities as evidence.

2:30 Well, I think this kind of approach, even though it is very simple, it enables you to do a lot of work with it. And actually, many of the current work on mathematical processes can be understood as specific analysis of facets of what we have in this kind of situation. if you get to know the rule by Mancosu, most of the chapters in there, I believe, can be understood as narrower aesthetics among aspects of the couple framework agent. Like, for instance, the role of visual thinking, which is, of course, related to cognitive abilities of the agents, in mathematics or diagrammatic elements or et cetera. But even if we just figure out these subtler aspects of mathematical practice, I think a very simplified model like the one I'm proposing also allows you to do some interesting analysis of the constitution of mathematical knowledge. Now, so we'll come to a second part. In order to do that, it is essential that we move on from the idea of mathematical practice in the singular to mathematical practices in the plural. It is a key thesis of my kind of approach that there are several different levels of knowledge or practice coexisting. This is again in contrast to Kitscher. In Kitscher's approach to the development of mathematical knowledge, you get the impression that there is something like normal science in Kuhn's theory, so that a given historical period in the development of mathematics is everything is regulated by a single paradigm, which is actually what Kitscher is calling the practice. In contrast to that, I claim, as fact actually, that different practices coexist historically at any given, well, at least in most periods that I know of, or maybe even know, historically several different practices coexist, and also within the single agent, individually, every

5:00 mathematician has knowledge of different practices. to make this a bit more concrete. A very simple example, the best one for an easy discussion of this, is an idea related with the concept of member, and we can talk about, for instance, counting practices. Everybody learns counting practices, actually, in our societies. Elementary arithmetic, that would be a different kind of practice with different symbolic means, with different characteristics from this point of view of the framework and also involving different kinds of cognitive abilities on the side of the agent and to give a third example related to this so we have counting practices elementary and say self-theoretic analysis of the structure of natural numbers that would be a third level again a different practice totally different language and so on. And they simply coexist within the individual and they also coexist historically. Or, to put another example, so that you can see that this could be applied to a totally different kind of example, think about one of the traditions in the 17th century, and think about practical geometry, Euclidean geometry, and Cartesian geometry. three different practices. And now my claim is that we have working knowledge, or mathematicians do have working knowledge of several different practices, and, and this is the most important element, we have knowledge of their systematic interconnections. Because it is actually this idea of the systematic interconnections between practices that allows me to do some work with this kind of approach. The inter-connections between different levels, cross-links, that be strictly invisible, guide the formation of new concepts, the development of new practices, and lead actually to the objectivity of new mathematical results. My approach is an approach to an explanation of the objectivity of Rukhjepo-Models. To give just another comment on this, this kind of strategy of analysis tends to go contrary to what is typical in foundational studies, because in foundational work there is a typical

7:30 drive towards systematic presentations of mathematics as a whole, under a single framework, so to say. And you can notice this even, for instance, in Quine's naturalistic ideas about these things. They still are driven by this kind of systematic perspective. And I simply urge that if we are interested in coming closer to some kind of cognitive analysis of mathematical knowledge, we have to resist the temptation of over-systematicity. Because it is actually the interplays that do the work. Now, third point. Let me say a few words about how to apply this to elementary mathematics and to advanced mathematics. In the case of elementary mathematics, we have something that is at least my belief, that is something rooted in our cognitive systems in the normal agent and her everyday practices. So here we have strong cognitive and practical roots of what we call elementary mathematics. I'm referring to things like practical geometry, so rural and complex geometry, or counting practices and things like that. I believe this is a strong commitment of practical roots that give a special epistemic nature to our knowledge of that kind of at the above. mathematics. And I am even indulging in talking about certainty in connection with such elementary layers of mathematics. But by contrasting that, I believe when we come to advanced mathematics, we have to consider, we need to incorporate into the scheme what I call a hypothetical of advanced mathematics. This is something I have elaborated a little bit both in Spanish and in English. But simply the idea is that we cannot present advanced mathematics simply as something that is constructed, so to say, step by step from the elementary layers of

10:00 mathematical knowledge, but rather, there is a central role for constitutive hypotheses in making possible the new practices, the new theories. And I will give you a couple of examples of this in a minute. So if I am right in this kind of viewpoint, there emerges the question about the interplay between the certain and the hypothetical in mathematics, and how can we still talk about some kind of objectivity of results. And this is actually the central point of what I am doing research on this moment. So, forth, a few more words about how to handle the question of the objectivity of mathematical knowledge. Consider, for instance, well, consider the evolution of mathematical practices having to do with numbers and with natural numbers, then consider, so consider that line counting practices, elementary arithmetic, set theoretic, the structure of the natural numbers. When we come to the more advanced viewpoint, the security viewpoint, there are some very actually very controversial hypotheses involved in that, like the assumption of actual infinities. After a talk on intuitionism, that should be clear enough. Now, the idea is the following. I simply accept the idea that the axiom of infinity is a problem. I think there is a lot of evidence for this and I am simply following on the footsteps of so many people in foundational work and 20th century mathematics that it quite safe to follow that line of approach. But my aim is the following, if one accepts, so to see how there is, how there can be objective results or highly non-arbitrary results about things of that kind, we have to consider precisely the interplay between

12:30 the new practice being developed. So, the idea of a set of natural numbers and the already accepted practices. In this case, for instance, the simplest example would be the following. If we accept the idea of a set of natural numbers, it is almost impossible, I believe, this needs to be perhaps to be argued a little bit, but I hope you will take this point. it's almost inevitable to admit the existence of any definable set of national numbers. So, on purpose, I'm not getting involved in arbitrary subsets or anything of that kind. It's simply definable subsets of the national numbers. Easy examples, the even numbers, or the prime numbers, or the set of multiples of any number you name, those are definable subsets and they didn't create any difficulties when this theory was being developed and there was all the debate about the answer of choice, but it was precisely because it didn't comply with this kind of approach of talking about infinite but definable Now, so my game is adopting a set of natural numbers gets you involved with the idea also of some, the adoption of some subsets of the natural numbers and previous arithmetical knowledge, on the basis of the already existing practices, you establish the result that there is a one-to-one correspondence between the full set and the subsets, the final subsets. So in this way you discover a feature, you are forced to admit the feature that in the an infinite set like the natural numbers, there is one-to-one correspondence with a proper part, what was Galileo's paradox and later, with Galileo, of course, it was a paradox because he was not really thinking in set theoretic terms, he was still thinking in the old framework of a theory of magnitude. But with that I can determine that part of the very definition of infinitesimum.

15:00 And I can assume that this is something to discover about something you have invented, in a sense. A hypothetical ingredient, but still in virtue of the interconnections with previous practices, there is some kind of discovery. So the traditional dichotomy invention discovery is purely inadequate to handle the conceptual subtleties of the mathematicians' way of working. Let me briefly say a few words about a more advanced example where we already have the idea of arbitrary subsets. So, in that sense, football set theory. This second example is the case of the power set axiom. And the idea here that I'm also elaborating in detail is about the very subtle and complex network of interconnections between this new hypothesis of Z-theory and previously well-established mathematical practices. This is an older step into the actual infinite, positing the existence of power domains, precisely because they are meant to be what people often call quasi-combinatorial domains which well of course there are critics who say quasi-combinatorialism has never been carefully defined and I have a tendency to agree with that but one feature is definitely clear about quasi-combinatorialism And it is the denial of the requirement that infinite sets be defined. There is a semi-constructivistic requirement about infinite sets, namely that they have to be definable, otherwise you can't handle infinities. And quasi-combinatorialism is nothing less than the denial of that requirement. And in that sense one talks about arbitrary subsets. Well, we are not necessitated to admit the possibility of quasi-combinatorial power domains,

17:30 we are not necessitated to admit the power-side axiom. But, my claim is, I will make two claims, actually, very quickly. That its adoption was not at all arbitrary, there was a strong motivation here. And second, that once the adoptive previous mathematical knowledge forces you to admit some results about that. So again, the links with other mathematical practices force you to some conclusions. So very briefly, I will simply sketch the argument. As to the motivations for power sets, there is an element that has not been, I think, sufficiently taken care of or thought about, which is the connections with the classical assumptions about the continuum and decimal expansions of real numbers. My idea is to be the following, since the introduction of decimal expansions, it was clear that real numbers within an interval of unident can be identified with all possible assignments of ciphers to the positions marked by the natural numbers. So, if you translate that using, so the idea that take a unit interval, the continuum of magnitudes in that interval leads you to all possible decimal expansions. And if you look at that, introducing slightly more advanced mathematical ideas from analysis as it was there by 1850, so decades before the introduction of some theory, this is basically this kind of idea, that you have all possible questions from the end to the ciphers, a set of ciphers, as given somehow. In details we are historically more delicate, but anything to simplify, this is already, by well-known arguments that you probably are familiar with,

20:00 this is essentially the same as having all subsets of the natural numbers the idea of arbitrary subset is already involved in here and then the second the second claim is simply that by well established arguments take either Cantor's first or second proof of the non-limitability of the continuum, both of them practices that are not specifically self-theoretical, they are pre-existing practices, and so again the link with previous practices forces you to conclude that the power set of n, and more generally the power set of any set, has a greater camminality than the set itself. This is an unarbitrally result by means of these interconnections with the practices that I'm talking about. And let's go first. Thank you. Yes, thank you for it. I have one remark and a brief question. So the remark is about the real numbers and the decimal expansions. Of course, it has, what is a degree with the classical or the intuitionistic conception of real numbers, right? Even if you abstract from that, it has at least been shown that the idea that to every real number corresponds a decimal expansion, that that is a fairly non-trivial matter. Yeah. From a few things that you have to deal with. So, even if you disagree with the intuitionists, it would at least show that it involves something quite non-trivial to uphold the idea that every real number has a decimal expansion, I think. but that was just a remark but my question was you've been speaking of practices now of course there are practices of all kinds of sorts, right? There are mathematical practices but all kinds of linguistic practices in sports what have you can you say something about what makes a practice a mathematical practice for you? Yeah, I do have some ideas about that can I elaborate on that quickly?

22:30 It's a video, I didn't bring notes that I have written on that topic actually. It is an open question for me right now, whether we need to... I could give some general ideas about that, but let me talk about a problem that I see in my reflections about how to frame the question of what is an analytical practice. One element I think is always involved, and I tend to seriously tend to teach a theory in a definition, is the symbolic elements that we, precisely, I felt the need of putting some symbols on the blackboard, precisely because typically when we are handling mathematical practices there are symbolic things involved. So that is one element that I'm highly tempted. And this sounds very contrary to Brauer's way of thinking about these things. But more generally, this would, in the end, link back with ideas about the mental. I guess I disagree with Brauer's basic philosophy about the mental. I tend to think these kind of elements are essentially what we call mental life and mental activities. But anyway, that would be a very complicated and very philosophical discussion. On a different line, on a different note, there are two aspects which are... So, mathematical practices are intellectual practices. So, that immediately gives you some restrictions. And they think that you have at least two components. Problem-solution and concept-formation, I would say. And those are two very interesting aspects to consider. But even by employing all of those elements and elaborating on that, it's still an open question for me, and I was beginning by saying that whether we have to pin it down by reference to some more concrete, so to say, elementary practices from which the rest elaborates, life, for instance, things having to do with counting and with measuring and with practical geometry. It will also influence, of course, when you're speaking of mathematical practices in the plural, and you're asking, as you alluded to a couple of times to the question,

25:00 are there systematic relations between those two practices? Of course, there are two. What would count as a systematic relation and what would also depend on an answer to your question? what what counts as mathematical yeah what doesn't yeah maybe if i well i would have something to say about your first comment would you regard the the practice of the physicist when he's doing mathematics in his own sloppy objectionable way as mathematical practice in other words would you you regard them as agents in your sense, possibly mutant agents? Well, I guess because they may be... There were things which are very, very objectionable from a mathematical point of view. Well, with the history of mathematics, their role has been very important. They have created mathematics which were not really mathematics, but nevertheless... Which sometimes became legitimated as mathematics, So classic example of course being hyperfunctions and Dirac's hyperfunctions. I don't have a ready-made answer to that. That should be an interesting question to consider. How does this kind of approach bear on the famous questions about the historical development of notions of rigor? Euler was very similar to many of our mathematical physicists today that people criticize heavily. But I don't have a really made answer for that. Anyway, I believe that an important part of the... I mean, in my opinion, there will be a physical number of acknowledging some of those practices and methodical practices, perhaps. The kind of framework I'm developing is actually very flexible and intentionally installed. Because I believe one can apply this kind of approach even to things that are not clearly frameworks that are not clearly theoretically developed. And even to mathematical practices like, for instance, Chinese mathematics 2,000 years ago, which are quite differently organized from what we are used to. But anyway, that's intentional. To make it easier, of course, it's a good idea to restrict to theoretical frameworks and then having very specific ideas about rigorous groups and so on.

27:30 That makes it much easier to handle, but I think it may be a good idea to keep it flexible and that's a way of destroying these people. Final. Please, we put the time aside. It ties to your last remark. What role would the notion of proof play in your conception of practice? Would you be liberal about it, or is there something distinctive about the proof that makes it convincing and mathematical, in the sense of mathematical proof and not something else. I mean, basically all of the work I'm doing myself is about mathematical practices that are proof-structured practices. And still, I mean, in reflections about the general framework, I'm tempted to think that it is a new idea to give it flexible enough so as to include mathematical practices that are not the only proof stretch. Like the Chinese example I was giving is exactly of that kind. So, I'm still very silent in mathematics. It's not simply blah blah blah or merely recipes or solving problems. There is justification of things in Chinese mathematics. I have it very much in my mind because we had a discussion about that a couple of weeks ago. But it's a very interesting example to consider for this, for the elaboration of the viewpoint. However, just as I said that in my current work I'm only considering normal agents, I'm only considering occidental mathematics. The next final speaker for this session is by the Center Social from the Autonomist University here in Madrid. The title of his talk is Natural Mathematics, a Pularistic Approach to Mathematical Prognation. Thanks for coming. I'm quite amazed by the title of this section. as formal methods formal methods

30:00 in the philosophy of science and the philosophy of mathematics I think this section has been named for formal methods going purple I will touch upon some of the issues that have been raised during the section from a quite different perspective that we can tie everything together during discussion. So, let's go. Accounts of mathematical recognition commonly divide into separate research fields, most notably philosophy and psychology, pivoting on a distinction that has been a much of great philosophical dispute, the distinction between personal and supersonal levels of explanation. I read it as one between accounts of mathematical performance and accounts of mathematical competence. This is a necessary distinction for complete characterization of mathematical cognition, both of mathematical knowledge and its acquisition. Thus, while personal level or philosophical accounts of mathematical performance requirements for mathematical knowledge, some personal level or psychologistic accounts of mathematical competence focus on the implementation or computational requirements for its acquisition. Now, the obvious question is, how do both levels of explanation relate to each other? One view is that both explanatory projects have distinct goals and thus are relatively independent from one another. Minimally, though, we want our philosophical and psychological accounts to be jointly consistent, since the former's explanation will usually be the latter's explanandum. But notice that this is a general requirement on any other theories we accept. Can we make a stronger, more substantive claim about the mutual relationship? In the course of this talk, I will examine two opposing views that not only claim the independence of the personal and subpersonal levels of explanation, but privilege one account over the other, autonomy theory and eliminativism.

32:30 I will point out some of the intrinsic problems and suggest how we might go about solving them by climbing out of the trenches and going for the panoramic view, in other words, by going for pluralism. Finally, I will mention some features, a characteristic of personal and sub-personal level accounts, and conclude by discussing a view that holds the strong interdependence. Let's consider them in turn. one of the main tenets of autonomy theories is the personal level phenomena or facility with mathematical concepts to reason about them and to apply those concepts cannot be reduced to the personal level facts and processes as the philosopher Jody Azouni points out this is so precisely because there's some personal machinations involved in our quiet facility such concepts aren't included in the consciously perceived contours of those concepts. Put differently, the push and pull processes that subtend our mathematical capacities don't seem to measure up to the normative requirements for rational, knowledge-yielding mathematical thought. However, by taking our applied facility with such concepts as the starting point, accounts of mathematical cognition in strictly personal terms usually come to an early end. The normative contours of mathematical concepts just force themselves up in us. But now the question is how. Autonomy theorists thus face the burden of explaining our share in the normative realm of mathematical abstracta, our acquisition of mathematical knowledge, without falling into some form of mysticism. Elimidantalism takes on this burden, but ends up throwing the baby out with the path water. Why not dispensing with personal-level talk altogether and replace it with hard-worn neurobiological facts? In the Greek for substituting the personal-level descriptions of causal and or functional

35:00 processes for personal level discourse, which they regard as utterly mysterious, eliminativists face the charge of changing the subject. For mathematics, no, for focusing exclusively on the subpersonal underpinnings of mathematical cognition, mathematics as a rational inquiry falls out of the picture. Are we ready to pay such a high price? While autonomy theorists devote themselves to the study of mathematical cognition from a strictly personal-level point of view at the cost of recruiting into mysticism, Illuminatists direct their attention to its neurofunctional foundations at the expense of losing sight of the normative requirements for mathematical thought. Let me give you an example. One important lesson to draw from current research literature on mathematical cognition is that the subpersonal processes underlying our mathematical abilities have quite a theory genius. For instance, even where a particular cognitive mechanism is dedicated to underwriting certain numerical abilities, it isn't the case that it underwrites all our mathematical capacities. As the neuroscientist Stanislas A.M. observes in his opening article to a special issue on numerical cognition, modularity is the fundamental concept which emerges from the present review. According to his proposed model, the idea of a unique number concept must give way to a fractionated set of numerical abilities among which faculties such as quantification, number constructing, calculation, or approximation may be isolated. Needless to say, it isn't only the enhanced model that fractionates our facility with numbers. Any model that accommodates the empirical results about learning difficulties, brain damage effects, uncompetencies, error effects, and response time experiments fractionates systems brought to bear on various mathematical tasks. It is part of our general methodological strategy that we analyze complex systems by breaking them down into simpler parts. However, if our concept of number

37:30 roughly encompasses those entities that participate in and resolve from, arithmetical operations such as addition, subtraction, multiplication, and division is a concept supposedly accessible to conscious reasoning as any other mathematical concept for that matter, then, as the Zuni notes, none of these results actually fractionate our concept of number. No more so as my concept of picking up a glass of water fractionated by the neurophysiological facts that show that the number of different and heterogeneous cognitive systems, none of which I'm actually aware of, are involved in my ability to implement that concept. If the worst comes to the worst, eliminativism ultimately leads to category errors, such as confusing the properties of our concept of number, whichever they are, with which we identify the personal level, with the properties of the subperson mechanisms that underpin our facility with that concept. So, both versions of what I have labeled as the independence view fought short of yielding a comprehensive account of mathematical cognition. Autonomy theorists fail to explain our acquired facility with mathematical concepts, while eliminativists miss the mark and fail to see mathematics as a rational activity. Let's turn our attention to a set of more promising positions and take a look at the interdependence view. But first, let me pause on some characteristic features of each type of account. Personal level approaches typically depict mathematics as a rational inquiry focusing on mathematical performance, on the contents of mathematical cognition, and on the principles that articulate these contents. Ultimately, this kind of strategy aims to shed light on the normative character of mathematics. Adapting an oft-created passage of John McDowell's invention, Artikel, Functionalism and Anomalous Monism, mathematical concepts have their proper home in explanations of a special sort, explanations in which things are made intelligible by being revealed to be as they rationally ought to be.

40:00 the hand, since all implies can, so personal-level approaches attempt to elucidate mathematical competence in light of the cognitive systems that happen to underpin our acquired facility with mathematical concepts. Note that this is an incredible question in which things are made intelligible by representing the coming into being as a particular instance of how things generally tend to happen. So far, this is not very telling. For one thing, given the way I've characterized mathematical cognition, it seems, in terms of knowledge, mathematical performance, and its acquisition, mathematical competence, it seems, you know, a trillion follows by definition that any comprehensive account of mathematical cognition will have to include both ingredients. remaining part of the talk, I will examine and critically assess a more substantive view of the relation between personal and sub-personal, as exemplified by Rochelle Goldman's and Rami Galisto's approach to mathematical cognition, and in particular, the account of the psychological foundations of arithmetic reasoning. there is substantial behavioral and neurophysiological evidence that the complex uniquely human, culture-specific and with medical skills exhibited by human adults rest on an internal number-specific system of representation known as the accumulator which is responsible for our ability to respond differentially to approximate number talk about the can be misleading in principle. The system can be fractioned into simpler subsystems, implemented in turn by a host of different and quite heterogeneous neural mechanisms. So nothing said so far is incompatible with the claims made earlier. So comparative and developmental studies indicate that the accumulator is an innate system with a long evolutionary history, for it is common because many species emerges early in human development and continues to function throughout the lifespan. The main source of motivation for the accumulator

42:30 comes from response time experiments and number comparison tasks. The data obtained from these experiments yields a robust pattern characterized by two distinct effects. The magnitude for science effect. That is, for fixed difference, the larger the numbers, the slower the response. It takes longer to tell 13 from 15 than to tell 3 from 5, even though the difference is the same. And the distance effect. For any two numbers, the smaller the difference, the slower the response. It takes longer to tell 7 from 8 than to tell 3 from 8. Both effects taken together conform to a single principle, namely that the response time and number compression tasks is a function of the ratio. This principle, known as Weber's Law, also applies to response data in comparison tasks of continuous physical properties, such as length, pitch, or duration, where discriminability of two magnitudes is a function of duration. These results suggest that the accumulator represents number using a system of continuous mental magnitudes. As each number is registered, a granite accumulates the way Mercury registers changes in temperature. The model can be readily understood by means of a simple and commonly used analogy. The idea is that the accumulator uses a mental measuring cup to gauge quantities. Imagine some liquid being poured into a beaker where the resulting level, a continuous variable, represents the total quantity. Put without the analogy, the model maintains that distinct levels of activation in a given population of neurons represent distinct quantities in the brain. The magnitude and distance effect can be traced back to the cumulative inaccuracy of the accumulator that renders numbers higher in magnitude and close in distance more difficult to discriminate. Indeed, the idea that the accumulator employs continuous variables instead of using discrete symbols to represent quantity, employs wide agreement in the literature. Yet, it is still unclear and it might have some dispute how to

45:00 interpret the system, in particular how to understand the notion of a mental magnitude. As we shall see, the list of Lengelman do it in a quite distinctive way. So, let them speak Mental magnitude, I cite, refers to an inferred, but one supposed potentially observable and measurable entity in the head that represents either numerosity or another magnitude, and that has the formal properties of a real number. When we mention a mental magnitude, they say, we mean an entity in the mind, brain, that functions within a system with the formal properties of the real number system. At the real number system, we assume that this system is a closed system. All of its combinatorial operations, when applied to any error of mental magnitudes, generate another mental magnitude. It is generally assumed that the positive integers form our basic system for representing and reasoning about exact number. In the literature, the invention of the reals is typically considered a far greater psychological and cultural achievement. Gallistow and Gelman turned this familiar picture upside down and suggested it is the system of real numbers that is the psychologically primitive system, both in the philogenetic and the ontogenetic sense. Our thesis is that this cultural creation of the real numbers was a platonic rediscovering of the underlying non-verbal system of arachnetic reasoning. The cultural history of the number concept is the history of our learning to talk coherently about a system of reasoning with real numbers that predates our ability to talk both phylogenetically and ontogenetically. According to Galisto and Galman, the reals are not only primitive, but also psychologically real. the real number system is literally instantiated in our brains. Long before we learned how to put them into words and started to represent them by means of natural language, the reals were already in town, lodged in our brains, as it were. The truth is attractive. It explains how the set of

47:30 mental processes can be rational and get realized in the brain by letting the physical properties of the system, the accumulator, mirror the normative properties of the target concepts, the reals. However, tantalizing as it is, in order to be respectable, and perfectly respectable anyway, the punitive mental magnitudes must validly enter into, at a minimum, mental addition, mental subtraction, and mental auditory. Otherwise, they do not function as numbers. This is the touchstone of Gallistel and Galman's startling hypothesis, and a requirement that they cheerfully acknowledge. Nevertheless, it is hard to see how the accumulator could instantiate the formal properties of the real number system, given that the main source for the accumulator comes from our ability to respond differentially to approximate numbers, a capacity shared by other animals or species, as representational resources may structure us rather coarse when compared to real numbers. Can the accumulator tell 2.43025826 from the same ending in 7? One could argue that it does, and insist that the mental magnitudes by which we reason are as fine-grained as the reals. What's the noise? Noise might infect the system in the process of storing, repeating, and comparing otherwise accurate measures, a response favored by Gillespie and his colleagues. On the other hand, the noise might be only apparent in that our experiments aren't yet sensitive enough to our fine-tuned representational capacities, a possibility briefly suggested by Lorentz and Margolis. Be that as it may, Galisto-Lengelman's interpretation of the accumulator cases a problem that, as far as I can see, cannot be overcome. The accumulator is ultimately limited in its representational power vis-à-vis the target system it is meant to represent.

50:00 Here's why. The reals are dense, that is, between any two arbitrary close real numbers. there is always another real number. However, there is no reason to suppose that for any two arbitrary close mental magnitudes, there is always another representable magnitude between the two. No matter how sensitive the accumulator actually is, there will always be magnitudes that form outside this discriminability threshold. Given these difficulties, the claim that mental magnitudes the proxies for the real numbers seems highly implausible, if not strictly false. It is more natural to think that language creates a representation of the reals when none was survived before. So, the account of arithmetic reasoning proposed by Deleuzebel and Gelman draws downward inferences from personal-level phenomena to the personal-level requirements for such phenomena. Now, these inferences are of a relatively outpowering kind, and relatively, since there is evidence for a system of continuous mental magnitude of representing number, albeit only approximately in the accumulator, in its normal classical interpretation. These inferences are of a relatively outpowering kind, I would say, and take the form of an explanation, a kind of transcendental deduction with a specific theoretical structure on the only game-in-town grounds. It is on these slippery grounds that they find themselves committed to the idea that in order to account for the conceptual structure of arithmetic reasoning, superficial processes have to mirror that structure syntactically. Although there is room for such a hypothesis, limited as we have seen, a truth must ultimately be empirically established. Why, in principle, couldn't we account for the same properties without the extra inward step? Why should we conclude from the fact that personal level descriptions have a certain structure that there is a matching structure in the brain? one might depart from Gilles de Lengelman's literal interpretation and argue

52:30 that the hypothesis of mental arithmetic is not about the brain as such. It is about the tacit knowledge of inferential rules, the laws of the arithmetic saying. But even if we grant that the structure of mathematical inference is not available to the thinker in conscious awareness as it is often the case, Why should it follow that it is encoded in the form of mental arithmetic? The move of letting the explicability of historical and sociological facts about mathematical performance turn on empirical facts about an inferential structure encoded in an internal arithmetic looks at best bizarre. One could nevertheless argue that this isn't empirically crazy after all. At one time, evidence about the age of fossil record implied that the classical theory of atomic structure had to be wrong. Chomsky, for instance, has stressed this kind of downward constraint on science for many years. However, it's one thing to have downward evidence that a certain theory sports. it's quite a matter to engage in a kind of reasoning that establishes a certain sort of theory on the sheer grounds that, as far as we can tell, there are no other alternatives. Well, we might recover an old Wittgensteinian theme as a plausible candidate and claim that all the relevant structure is found in the experiential practices of the mathematical community. The idea being that we lay down rules, a technique for a game, and then follow the rules, as Wittgenstein once said, so that the rules are the touchstones used to indicate when we have made mistakes and when we haven't, when we've operated in accord to them and when not. Now, as a personal level requirement for following a rule weaker than the possession of an internal mental arithmetic is to have a minimum capacity or a psychological disposition to mimic some of the results of its application, which allows for varying degrees of success and failure

55:00 and in turn enables us to see mathematics as a normative inquiry. Thank you very much. Thank you. Yeah, I have to say, I think there seems to be a quite incredible mass of conceptual confusion in this body of work that you report. If the claim is being made, and I took down word for word what you said, if this is correctly reported, if the claim is that the system of real numbers, by which I tend to mean the a system of real numbers as axiomatized, you know, indedicate in terms of ratios and building in cuts, is phylogenetically and ontogenetically more primitive than our representation of the indigators, then that claim just seems to me to be completely incoherent. However, if the claim is that some representation of extensive quantities of the kind which was the subject matter of eudoxys is very this. Some representation of extensive quantities of the kind which are ratios of magnitudes which are grasped in geometric or kinetic intuition is final genetically or non-genetically more primitive than our representation of intuition. This I can understand, but it seems to be a colossal conceptual error to represent this as saying that the reals are more primitive than the individuals in the world. That's why I have cited them. Okay, so my question is that, that was a remark, my question is a quick one. Has this body of work in neurocognition pinpointed any basis for the distinction between the representation of Kahn or Norton or Nandas? The way they do it, I mean, what they really try to do is to find out how the data, the response time experiment, and so on, can be fitted by doing this kind of

57:30 fundamental reasoning. And they say, well, you cannot, given the way we estimate quantities. When you talk about the estimation of quantities, if you spoke about, you know, the distinguishing between 7 and 8, the distinguishing between 3 and 8, they're We're obviously talking about distinguishing between three-membered and seven- and eight-membered collections. No, empirically, given objects. I don't know what they use. And that's what they're talking about in that context. So that appears to relate to carbon. Yeah, that's why they talk in that context, when the cognitive system is working online, if you want. But when we are reasoning offline about number, we still draw on these cognitive mechanisms. And the idea is, they would claim, well, it's subtly mysterious how we came to the concept of real number if we don't assume that it's already innate, in a way. instantiated in the workings of a cognitive system, the accumulator, in this case. I find it bizarre. I think my dad is going to protest that they see the system talking about the real numbers in this context rather than about extensive magnitudes. Okay, I'm sorry. I'm probably good enough. Any other questions? I was wondering from the beginning how would you like to account for the so-called personal level? And of course the suggestion at the end may come to the path. Would you be tempted to think that the personal level is intimately linked with community games or community practices or language? which I think it is, but I think one has to go beyond my last sketchy remarks. It's not so easy, well, we are all familiar with the rule-following problems and Wittgenstein's remarks on rule-following

1:00:00 And it's not so fair that you solve that by saying no, there is an ending disposition to follow a rule. But I think that we have to keep the normal agent in mind here talking about and actually look into the psychological experiments about how we actually, how brains actually manage to grasp rule, for instance. And I would suggest that, yeah, that language plays an important role, and I would go that line, keeping an eye to the empirical stuff of psychology, and looking whether it's actually empirically the case that we follow a rule by such and such. but it's hard to on the one hand one is drawn to be an autonomy the theorists will never be solved at the empirical or super personal level we're talking about different and I'm quite sympathetic to that but on the other hand it seems that we have to get a clear what we are and how we work and for that we went to a little bit of psychological work and criticized it and say look it's a crazy hypothesis that the reals are somewhere instantiated in Ukraine they claim that they're potentially observable All right. OK, I can conclude. Thank you. In the session, we will be using general relativity with no pathological constants and general relativity with pathological constants. What the standard use of the answer to my problem says is that we cannot distinguish between

1:02:30 these three theories because they have the same influence and this is a consequence of having a definition of invariance rule that depends only on the symmetries of the absolute But we can observe that in general relativity we talk cosmological constant and there are some degrees of theorem that we are not taking in some time. And this is because the theory is a scale improvement, so we can fix, I mean we have this absolute object which fixes the value of the scalar density in a neighborhood and this fixes the volume element after a constant the theory is a skeleton and so this we have this extra degree of freedom in the value of the volume element so intuitively This extra decrease of phenom should be taken from one computer, the influence group, and they should enlarge the influence group. And we should say that now the influence group is not the volume resulting in the film But these two products, the different values that this volume element can take, and this is isomorphic to the difference. So, by making this contrast between general relativity and minimal relativity, what I is that what I call the standard use of the Anderson treatment program gives you an anti-intuitive result by saying that with various groups of modular relativity and general relativity with a cosmological constant have the same effect. So, this might be a reason to try to question with the logic of the standard use of the Nelson Friedman program, but, I mean, related to this one, I think that there might be other

1:05:00 reason, we can formulate this reason in a different way, and we can think that it is a strange definition of the universe that depends only on the formality of the absolute subjects. We can, maybe it's useful if you try to think of two different theories where where in both of them we have the same absolute object, but they enter into the theory in a different way. The others of the program would say that the invariant group for the two theories is the same, and maybe what one says about the invariant group of the theory should be more dependent on the dynamics or the way an absolute object relates to the other. So, this doesn't seem to be captured by, and there's a definition of invariance in some cases. And another concern is whether the use of a local definition of absolute objects may leave a global decrease of freedom that one wants to count as belonging to imperialism. So I think the assumed equivalence between background independence and lack of absolute objects is not necessarily true and one could try to use alternative definitions of imperialism which, to modify the original Anderson's definition, and here I propose one, as I should say, to define the difference group as the simple of the distribution group, such that for all the transformations belong into this group. It takes you from solutions into solutions by changing only the, applying the transformations only to the absolute objects and leaving the other objects and changes.

1:07:30 And here I also propose that maybe this definition of experience as the extra advantage of providing a way of explaining what I said here, the acting part of the metaphor I was talking by trying to relate the invariance group to the degree of acting of the absolute subjects. So, in order to define the invariance group, it's not only important whether the theory has or doesn't have absolute subjects, so also, well, to what extent these objects act on the others, on the other technology. This idea because the bigger, the advanced group is the more transformations that change the absolute objects without changing the other objects, so we could say this means the less the absolute objects. So, just to conclude, I'll summarize, I'll discuss, I agree that the definition of local absolute object leaves space for degrees of freedom that might be relevant for the variance group. The definition of the variance group that includes these degrees of freedom could break a equivalence between local independence and the lack of actual subjects in terms of gender use, and a definition of local independence in terms of, in various groups, is free, or maybe free, or most of the problems that you would like to ask. Thank you. Questions? OK, I just want to go back to the counterexample that, well, you did propose, somebody proposed. Now, you said that, yeah, you said that we can do this, this square root of minus g equal to 1.

1:10:00 We can do this about any metric. We can define a local coordinate system. Is that right? where, by defining the coordinate system, we can arrive at this. And you said that this, in some way, reduces the symmetry group, or the invariant group. But compared to what? I mean, the only thing we've done here is introduce a local coordinate system. So what exactly, I mean, when you say reduces it, And I wonder if... What I mean is that if one assumes that the invariant group which is a normal activity should be the whole different division group, and you find out that this is an absolute subject, then, using Anderson's definition, the invariant group is the symmetry group of these objects. It's reduced in the sense that it's more of good than the whole thing. Okay, but surely this is only an absolute object within that local coordinate system that we've just defined. So rather than saying that it's introducing and somehow there is an absolute object there, we've defined a local coordinate system which allows us to see the metric, or which sees the metric, that local system sees it as if it had some kind of absolute object component to it. I mean, I know that you can... Yes, this is the representation of absolute object. I mean, it's local, it's defining the equivalence between this object in two objects in order to be equal as absolute. You should match one with the other in the neighborhoods of 80 points. So it's a local, a globally local definition. Locally global. I have a kind of relationship. Oh, sorry. I didn't. Dennis, you go first. No, no, no.

1:12:30 Well, mine is directly related, so you go first. Okay, well, then, yeah. Can you spit the slide that you're playing So, what is A.J., or G.R. without a cosmological concept? Yes, I suppose to use the problem that one should take a population to a number of activities where both the natural intensity and the scalar density are available. You can write it on a random form. So, AHA is the scalar density. So, is this really going to be true? Do you want the various groups in that case to be? I think this is true. I think this is true that we have 30% in the human systems. So, you are trying to imagine whether some, for some, if the definition is not true. I mean, if you're leaving the massive fields, they are changing them.

1:15:00 I just wondered, is it a necessary condition, or is it going to be the invariance group? of the absolute object probability to be a subgroup of the different morphism group. Because if you have an extra symmetry, if you have a theory with an extra symmetry in addition to a different morphism, say you have an additional IC2 symmetry, then you could say, I can have an absolute abstract object with respect to this new symmetry group, namely exactly by this definition, but it doesn't have to be an absolute object with respect to a group that is a subgroup of the different morphism group. Would that also work, or why, why not? So, thank you. So, you are wondering what I think for you, and you don't demand that the audience group is a separate or a different audience group, or that the definition is a separate or a different audience group. Thank you very much. And I would like to thank all the speakers.