Theory & Object (Session 6)

Hello and welcome to the sixth session of Theory and Object. I'm going to pass the mic to the course instructor now. Thank you very much Theo. Thanks everyone. So as promised, I'm going to wrap up the Stegmuller this session, give an example about the acidification projects formalizing the structure of scientific theories by way of the classical collision particle mechanics. Once we do that then we basically go into depths of the nightmare which is the work of Rudolf

Carnap and for the I would say the next three courses we will just concentrate on the work of Carnap which is both fascinating and extremely demanding and after that you know we will take the other sessions. So anyway, shoot if you have any questions, comments, suggestions, so on and so forth. Hello? Hello, how are you? I have a terrible headache, so I'm in the bed, so I won't switch

on my video. Don't worry, don't worry, don't worry. So, but I just wanted to say that one thing I would like to know because I was reading like the summary that you shared and I'm a little bit behind in reading so I'm very much interested in this different kind of models like partial potential yes potential and I would like just yeah I was really like some examples like because it seems to be very important and interesting. Yes, so I will basically, as I mentioned, the reason that I want to make that basically example

via classical collision particle mechanics precisely because it allows us to first grasp very, kind of a simplistic way, the difference between M and P and MPP, model, partial model, partial potential model. Not only that, but also what do we mean by the logical formalization of the structure of theories and what is exactly we can gain from it? So I will definitely make this example and I have taken some notes which I will do a screen sharing so you can see the formulas as I read them.

But also, Leanna, have you read Road to the Structure? There is a chapter in that book, it's a collection of Cohen's essay. Yes, I have actually this book in Russian. Oh, okay, great. So there is a chapter in that where Kuhn actually really in the most simplistic terms elaborates the differences between these three terms, model, MMP and MPP. Do you know the chapter or I will just look it up when I get home?

You just can't search it but I will I will I can't sorry I can't remember it correctly but I think it's at the middle of the book but I will definitely if you want me I will I will search it and I will give you the back page number. worry I'm capable of doing that myself my apologies I think look it has Laura been in our class Laura have you been in our class or is this your first session I've been here my apologies so that's you know whenever I say something like that you should take it with a grain of salt i have unset dementia

uh i missed a couple there were a couple where i wasn't here but i've been here for the majority okay okay i see i see so is there any suggestion question or comment that you can come up with um not right now but once we start talking probably Okay, great, excellent. So, let's start. So, I remember that Theo asked me the difference between a statement view and non-statement view,

view, and now that having very briefly looked at the kind of formalized apparatus behind a singular sneak paradigm of formalizing scientific structures and dynamics, let's look at the difference between a statement view and non-statement view. and I if I remember theory was that you who posed that question yeah so

Essentially, when you are thinking about the relation between Kuhn, Sneed and Stegmuller, can think about that a signal or an estate try to form a formal approach to scientific theories Kohn in his original work the structure of scientific revolutions tries to give a historical approach to a course of scientific

evolution then a signaler goes and uses a sneak formalization the so-called signification of scientific and dynamic of a structure of theories so as to show that in fact Cohn's historical approach to history of science can be thought logically and I mentioned what would be the advantage of this logical perspective logical a structuralist perspective fire up end accepts and recognizes a signal errors as notification of the

Aquarian paradigm but nevertheless goes on and criticize this in order to refine it. So let me give you a very brief again account before I move to more detailed accounts. Traditionally we have two views of the structure of scientific theories. What is called the statement view and what is called non-statement view. The example of a statement view would be early Carnap in Aufbau, the logical structure of the world. According to the statement view,

a scientific theory is a set of statements. Exactly what kind of statements? Well, they can be theoretical statements but such statements bottom out in what Carnap calls atomic facts. What are exactly atomic facts? These are what you might call to be statements about our observations. As we will look into the work of Carnap, particularly the early work, we will notice what exactly how this can happen. That you have observations and then you have corresponding sentences to such rudimentary

observation, like saying that this rose is red. Red, this is red, the assertion that this is red is an atomic fact or a statement. This statement is not written essentially in natural language. It can actually be translated into a set theoretic number whose members are real or natural numbers. For example, you can assign numbers the color red. Of course, the collection of such statements or atomic facts creates greater collections

set of statements and so on so forth. So if a scientific theory is a set of statements according to this view, there is empirical vocabulary and theoretical vocabulary. Empirical vocabularies pertain to observable things like for example this rose is red as theoretical vocabulary pertain to unobservable things unobservable things there are some correspondence rules that allow us to reduce the theoretical vocabulary

to the empirical vocabulary. This is the core tenets of what you might call to be logical empiricism, which we will look at very closely in the next few sessions in the early work of Carnard. Essentially, this is what you might call to be the doctrine of logical positivism or logical empiricism, in the sense that it takes theoretical vocabularies to be reducible to base empirical vocabularies which are couched in set theoretical terms and whose variables are numbers rather than names couched in terms of natural language.

Now, so this was the first formal approach, beginning at least since the time of Carnap. The second formal approach is the one that is advanced by Sneed and Stegmuller, among other structuralist philosophers of science. According to the non-statement view, a scientific theory is a pair K-I, is a tuple K-I, that consists of a structure core K and a class of intended applications I. K in this tuple

is a class of semantic structures which is axiomatically defined. And I mentioned, you know about this whole consequence and significance of axiomatization of theoretical systems. So K is a class of semantic structures which is axiomatically defined. I is a class of fragments or substructures of elements of K and I mentioned that I stands for intended applications. I is in fact what you might call to be representing the empirical world. So to answer Theo's question, the difference between a statement and non-statement view

is not about formalism, they are both formal approaches. The difference comes into the foreground only by virtue of whether your theoretical statements can be reduced to atomic empirical facts or they can't. If they can't, then we are talking about the intended applications of a certain theoretical structure, like Newtonian laws. And in that case, it is the intended application that represents the empirical world.

So essentially, the difference between a statement and non-statement view is a difference between whether we take theoretical statements to be reducible to the empirical facts or not. If we take them as reducible, then we are in a statement view. If not, then we are in a non-statement view. Questions? I might be missing some subtext here, but is this kind of like the distinction that Kuhn makes between, say, looking at a theory reducible to the set of empirical facts versus the structuralist view of theories,

whereby the empirical observations are encapsulated in the theoretical view by way of the structural approach in theory? Yes, but in correction to your second, so everything you said was correct. I would say that the non-statement view essentially sees the empirical purchase of a theory not in the reducibility of the theoretical statements to non-theoretical, namely empirical facts, but in terms of the dynamicity, the dynamics afforded by the structure, the logical structure of the said theory. Intended application is really the range of the dynamicity of the set theory.

And it's only intended application that involves empirical facts, not the theory itself, not the structure of the theory itself. yeah that's roughly all the ones for that I was thinking thank you so the statement view accordingly accordingly the statement view and the non-estatement view are not theories about fields but different forms of formal notation for

theories. In one formal framework it is possible to in fact reduce theoretical statements to empirical observational atomic sentence or statements. In the other it is not possible. So we have a question of different formal frameworks here and not a question of different scientific views that I would want you to pay attention to. So when we are talking about a statement and not a statement view, we are not talking about different scientific views. We are simply talking about different formal frameworks and what they afford us, what they demand us to do in one framework.

have to do the empirical reduction, reduce theoretical vocabularies to base empirical vocabularies, in the other we cannot do that. And essentially we see empirical vocabulary simply as the dynamics or the range of intended applications affording by a logical, a structural core of a theory. This might be a really naive question, but then, so we're talking about various authors, Sneed, Stegmuller, etc. Who would you ascribe to the statement view and who would you ascribe to the non-statement view? Well, a statement view is Karnoff, differently. Non-statement view is differently a Sneed and Stegmuller.

And Kuhn, actually, later Kuhn. Non-statement view is Sneed, Leitkuhn, and Stegmuller? Yes. Yes. Okay. And then Theo adds NSUPS? Yes. Merza, a following question, like, does distinction statement and non-statement formal like views how does it map to distinction let's say because I read some stuff like syntactic or semantic also like there's also a pragmatic kind of approaches to yes how do they map to this

kind of distinctions well you see to be honest with you I don't think that it's a clear it doesn't clearly map but if I want to be a little bit more I would say broaden the scope of how the distinction between a statement view and not a statement view map into the question of syntax and semantics you can say that the a statement view which is that of corner is essentially syntactic precisely because syntax allows us to do that labor of reducibility of

theoretical vocabulary to empirical based vocabulary those atomic statements couched in terms of set theoretic predicates. Now of course the reason that I said that we can in a very clear cut map this into syntax semantic issue is precisely because in logical structure of the world of Bao, Karna espouses this kind of syntactical view that I just mentioned. But in the logical syntax of language, after having failed in the project of logical structure,

logical empiricism, after having failed in the project of logical empiricism, Carnap sees syntax no longer as this kind of, that should be defined in terms of reducibility of theoretical statements to empirical sentences. Instead, he says syntax, what he calls semantic in disguise. So essentially, later on, Carnap is not pure formalist anymore. his formalism is not really a consequence of him being against the idea of semantics. It's just that he thinks that prioritizing the question of semantics over syntax leads

to different forms of psychologism, particularly, for example, in the Kantian framework of representation, representation, where you can say that the function of language is to represent the structure of the rule. Karnap no longer thinks such things in logical syntax. He thinks that language does not in fact have a representation of function. It is purely a structuring function. And to that end, he sees language essentially as a syntactic combinatorial system in which symbol, in which you can come up with any kind of symbol, design an inscription or symbol.

And then as long as the symbol ascends in explicit relations, grammatical relations to other symbols then you can call it a language and this lang within this language essentially we are no longer doing any representational job in the sense that we can say that you know behind every statement that we put forward in this system of language there is an empirical atomic fact So he no longer thinks about this representational value or correspondence between theoretical sentences and empirical sentences.

So this is one. The thing with a Stakemoller and a Sneed is that I have noticed that, again, not probably their earlier works, but as they matured, is that for them the question of semantics is extremely important. Syntax as well. You know, obviously syntax gives us the resource for systematic formalization. But essentially every formalization should be semantically interpreted in the last instance. This is what you might call to be the labor of theorization. So, a signaler and a sneet in this sense have very much in common with POTNAM, precisely

because POTNAM also thinks that theory cannot just be talked about in terms of pure syntax. You might do syntactic manipulation, syntactic combinatorial formalism, so on and so forth, but at the end of the day, in order for you to understand what you have been actually doing in the framework of it of a given theory you need to resort to semantic interpretation of course the question of pragmatics makes things even more complex it depends on what kind of pragmatism actually are we talking about well I see pragmatism

at the end of the day as an interface between syntax and semantics in the sense that it's it's the bridge between form syntactical form and semantic content or semantic interpretation or significance and of course I don't think that any of these philosophers of science have actually given you know a kind of robust attention to the question of pragmatics namely the interface between syntax and semantics they have only talked about inter subjectivity so on so forth but these are just really vague ideas to be honest with you only

in the wake of advances in contemporary logic and computation that the question of pragmatics as a bridge between syntax and semantics can be coherently tackled with. Everything else, even Perth's early pragmatist to, you know, Brando, who is actually, I would consider him, a very, very sophisticated pragmatist, in so far as they don't really understand how syntax can become the imminent point of genesis for semantics, the way that they question

pragmatism usually, not always, usually ends up in some soap opera of intersubjectivity, sociality so on and so forth but pragmatism is actually the formal condition of bridging syntax to semantics form to meaning or content for pragmatics is it also a way of finding out when you implement or apply a statement view or non-statement view I'm not sure about that I mean I need to really think about this it can be it can be precisely

because when the issue of pragmatism is raised the issue of pragmatism is forced first and foremost is a contact context sensitivity of application of forms or syntactical forms the kind of that is both common to the non statement view and statement view so in that case yes Yes, I would say yes, precisely because it brings the issue of context sensitivity of what kind of formal approach we should apply to a certain given system, theoretical system. I mean, I think that's, if I'm understanding the paper correctly, at the end of, is it chapter one, where Stegmuller says, we thereby touch on the fundamental question of what

distinguishes a theory of mathematical physics from a mere mathematical theory. Yes, yes, yes. Of course, you should understand, Theo, that in the canon of analytic pragmatism, the statement view itself is fundamentally impossible. Why? Precisely because according to the pragmatist doctrine you cannot reduce a theoretical vocabulary to a base empirical vocabulary. Such a reduction requires the intervention of modalities,

modal vocabularies and hence it's not pure reducibility of the kind of vision that Carnap had in mind when he wrote off-bowl. You can, you can, those of you who are interested in this pragmatist issue and the issue of reducibility, I mean both in the terms of, in terms of science but more generally in terms of logical vocabulary to one another you can look at the Brandom's work on modal vocabularies he has an essay I cannot remember the title but you can also look at I think it's the second chapter of between sayings and doings which is a

Masterpiece on Analytic Pragmatism. Robert Brando. I mean, my sense of the Stegmuller paper is that he is trying to um, I don't know if it's, would be like, go about describing why the statement view requires a non-statement or structuralist approach. Yes. I think a non-statement view is far more closer to the spirit of analytic pragmatism

than the statement view. I mean, that's my understanding of it, too. It's just that I think he points out the problem of the statement view, and I guess maybe, I don't know Karnap's work well enough, but it's like Pernup identifies why it's a problem. And then, but we also can't wholly get away from the statement view. It seems like we can't just have a non-statement. No, you see, non-statement view does not exclude the statement view. It's just to say that the idea of reducibility of theoretical vocabularies to empirical atomic facts or empirical vocabularies is not going to happen. Such a one-to-one correspondence is destined to fail.

And we will look at Carnap's work. Why is it failed? Carnap himself admits when he writes logical syntax of language. As I mentioned, literally, Carnap's logical empiricism, them. You know, I mentioned here that there is this famous saying by a student of Carnap, that Carnap was this intellectual tank, polite intellectual tank, who rolls over all. He was literally, he was one of the most genius philosophers of all time. And he's under recognizance, of course, because, you know, people think, oh, well, this guy was a positivist.

But those people who actually talk about positivism, they really do not know what positivism was. It was far more revolutionary than any philosophy that had come before. So Carnap made sure that if he's going to explore the idea of logical empiricism or positivism, it should be so stern and it should be so thorough that if he does, in fact, fail, then that would be the failure of the hypothesis and then that would be the end of it and he did and he admitted it so Karna I don't think that the non statement view exclude a statement view it's just that a statement view commits

to a very straightforward kind of formalism that is not tenable in understanding the structure and dynamics of scientific theories, or science in general, precisely because there is no such a thing as a one-to-one map, as an isomorphism between theoretical statements and empirical statements, to which the former can be reduced. So you say there is no isomorphism, but I mean, do you believe that you still can kind

of trace the relationship? Yes. Which is probably very much more complicated than just an isomorphism. Absolutely, absolutely. Precisely because the reason I'm saying that it's not a one-to-one mapping or isomorphism is precisely because such a reduction, such an empirical reduction, requires the intervention of other kinds of vocabularies which are not empirical, like modal vocabularies, nomological vocabulary, so on and so forth. Permit me to switch roles with Theo playing the typical skeptic. He's a rationalist skeptic. He's not just any kind of a skeptic.

No, no, no. He's one of the best I've ever seen. But what about the other inverse relationship, not just the reduction of theoretical statements to empirical statements? but what about like the reduction of empirical statements to be accountable by theoretical or you know I mean essentially that's not a reduction not a reduction yeah because you see empiricism takes itself to be the data where a theory takes itself to be a structure what is a structure a structure is what formats the data so when we are moving from the datum to

our structure we are not doing reduction you are actually doing enrichment and complexification and reformatting of that data that we are talking about so essentially what we have been trying to understand very implicitly is a relation within a structure as that which formats data and empirical data in particular and that was such empirical data the relation between these two essentially what you might call to be has led the group all the great battles in philosophy of science but also philosophy in general it's it's really

the fight between empiricism and rationalism if you want to kind of over expand it should we have a break Theo and then come back yeah let's take a is it a minute break okay for or we can have it shorter if you want. Let's dig steadfast into Stegmoor, perhaps. All right. Ten minutes in.

Thank you. Thank you.

Thank you. Does anybody happen to have any on-topic small talk in this interim period by any chance?

hey what's up man can you hear me yeah i can hear just fine yeah do speak um i didn't want to bring it up because it was kind of off topic from getting back to stegmuller but Reza had mentioned to take a look at the intro and the other section 66 and whatever of Aqbal. And in doing that, it actually led me down an interesting road to the other guy, Goodman, which Reza had mentioned previously. I think it was Ways of Will. But they start to get into this weird relationship.

and and carna describes it i feel like and maybe goodman is even more articulate of like not of how to avoid idealism and realism and assume neither and and he he takes a constructivist position and i relate it here because we were sort of trying to relate the pragmatism into this question of statement and non-statement view and his constructivist position to me also i don't i don't know if isn't making sense this i just was reading this this morning but his construct his constructive position was basically seeking to also not not assume like he was saying that

early pragmatists fall into this naive realist assumptions and i don't i don't know if that's traction if that's completely true but he was sort of trying to sit back from from either a real materialist position and and allow these things to be constructed similar to very actually very similar i think to what reza has been talking about when he presents his views over the last couple weeks so what was the final point i'm sorry i missed that i i guess i was just pointing attention to the and the way that it continues the earlier discussion of trying to sort of relate pragmatism into this discussion um in his his and then constructivist um alternative

Yeah, no, I see what you're saying. I mean, I haven't read any of those, you know, not yet anyway. You know, I've been somewhat slightly busy. But, I mean, what you said, you know, with respect to the sort of author you met maybe you said could mean with like not quite the realism view but also not the idealism view and taking more of a constructive approach you know um i mean i think that that probably makes like quite a deal of sense i mean

I mean, for a number of reasons, but, you know, like the fact that you cannot adequately appeal to some sort of like externally given what exists sort of out there, you know, without taking, you know, your own sort of subjective way to sort of format that understanding. But conversely also, you know, if you have some sort of, you know, sort of essentialist dogmatic way of sort of imposing what types of things you take to be out there, you know, then you also fall into a similar but different kind of trap.

And looking at these different things as being able to sort of like test each other and be able to interact with each other, I find very important. And with this type of thing, then you can, I think that you can probably, you know, avoid the two traps while maintaining a certain kind of certainty, but also a certain type of like critical self skepticism with such a constructive type of approach. In response to Theo, he wrote on the sidebar that he's still wrestling with the wrestling with notion of structure without object. You see, the thing is that a structure is just

simply a name for theory. A structure and theory are co-constitutive. Now the co-constitutivity of object and theory, or object and structure, or being a structure, to use Lorenz Fontel's term is that it can be approached in different ways one in the tradition of the statement view a la Karna in which the structure of your set theory can be reduced to atomic facts empirical facts this even Karna himself says that is impossible in his later age in a sense that you would say that even the most

staunch logical empiricist like carna admits to the fact that empiricism is not the groundwork of doing science is not really a medium by which we do science it's just rationalism but rationalism and that brings the complex co-constitutivity between theory and object or a structure and being in the sense that we do do a structure we do logical stuff that's the mathematical or logical pure mathematical logical but then

Then we create the core of our theory, okay, according to a semblance of how we have already observed the world, we have gone and about to explore the world empirically from our own perspective of our experience, not empirically, from the point of our experiences, or manifest image in a Solarzian way. And the thing is that, of course, this theory should conform to some semblance of empirical facts. But these empirical facts are not the core of our set theory.

It is what you might call to be, what? A range of its intended applications. That is in the Kuhnian sense, so you said that Kuhn gave a very kind of unconvincing account of science, but this is the whole point, that you make the theory via applied mathematics or mathematical physics in these fundamental areas, then of course your theory should be pruned by its range of intended applications. Once we move to the range of intended applications I, we can do a lot, we can do observations, we can might in fact find what one calls

empirical anomalies, as long as they do not compromise the theoretical structure of our theory we are in okay way we don't need to change anything but once they accumulate and they do in fact compromise our theoretical core then we see that either our structure our theoretical structure is not capable of affording the object, empirical object, or it is that the empirical object that we are seeing might hypothetically be the fruit of a potential new structure.

And of course that leads us to found and construct a new theory. Rationalism by no means ever says that empiricism is wrong. All rationalism will tell you, so as a structuralism in this sense, is that the structure itself, the theory itself cannot be simply reduced to empirical facts or atomic statements.

That would be just a variation of the myth of the given according to Salah's. We always begin from what Pantel says, the dimension of a structure, and by dimension of a structure, Lawrence Pantel said, means mind, mind in a classical Kantian sense. We always begin with mind. Mind is that which gives structure to the datum, the empirical datum. If we take that the datum itself gives a structure to us, then we are in the business of pre-critical

philosophy. It's Lorenz Puntel, L-O-R-N-Z-P-U-N-T-E-L. And the name of the book is A Structure and Being. It's one of the best reassessments of philosophy of mind, philosophy of logic, and what you might call to be theoretical structures. It's very, very comprehensive in the sense that it really covers a sweeping ground from the time of Plato to Kant, Hegel, Sellars, and Brandon, and even Karna, and tries to

really encapsulate what is exactly what we mean by a structure and why is that a structure is the most important dimension of the mind. I mean, you see, transcendental, essentially what you might call to be non-statement view is a child of transcendental philosophy. Why is that I'm saying that? Because Kant as the prophet, initial prophet of the transcendental philosophy was the first philosopher, maybe not philosopher, I would say that Plato was the first, but let's not get to quibbling about these historical facts.

reason that it says so is precisely because you see pre transcendental philosophy the mind the dimension of a structure is a blanket slate is a tabula rasa and the data whether sense data or the world as such or reality an external reality is the structure and of course this is what Sellars calls the myth of the given where you see the world as this stamp that imprints its own structure on the mind which is like a tabula rasa a blank slate post-cant the whole situation

is getting the other way is reversed their mind is data is their structuring data and the world is a tabula rasa so allow me to move forward So, having briefly talked about the difference between a statement and not a statement view,

in the sense that they are both formalism, but just that they're, the kind of formal approach that they engage with are fundamentally different. We should say, along with Cohen, that a scientific theory is not merely a formal thing, precisely because the scientific theory has psychological, historical, and sociopolitical or sociological traits or characteristics. If we analyze not only formal caricatures of scientific theories but theories in their historical context, we obtain substantial changes of our picture of scientific theories. Theories,

as Cohen says, are paradigms that are shared by a whole scientific community. In normal science, what Cohn calls normal science, the scientific community is concerned only with puzzle solving in the realm of fixed scientific paradigm. Failure of an experiment does not lead to the immediate rejection of the whole paradigm. The empirical vocabulary of a scientific theory is theory-laden always and all the time because it is constituted by the theoretical paradigm that the observer or the scientific community holds. A paradigm forms a scientific worldview of a highly unique nature including formal psychological

and methodological assumptions. So with that said, now we have a very clear view of the Stegmuller-Sneed project in philosophy of science also having briefly looked at the axiomatization or the axiomatized axiomatization of scientific theories now let us look at a very concrete example see how we can logically talk about a given scientific theory with

everything that we have talked about so far Our first example is one of the simplest real-life theories to be found in physics, collision mechanics. Historically, collision mechanics was also one of the first physical theories to be treated in a quasi-axiomatic mathematical way.

The basic concepts and laws were already developed within the Cartesian program for the physical sciences in the middle of the 17th century, before the advent of, in fact, Newtonian mechanics. Frequently, if collision mechanics is treated in a standard expositions at all, this is done within the framework of Newtonian mechanics, so to say, as a sub-theory of the latter. This might have some practical justification, but for conceptual as well as historical reasons, we think it is more adequate to reconstruct collision mechanics as a theory by itself

before newtonian mechanics so in in essence the kind of example that i'm going to elaborate on we are not even going to look at newtonian make a theory of newtonian mechanics but simply look at the implicit theory in which collision mechanics is constructed. Once we do that, we see that the logical structure of such implicit theory already encapsulates Newtonian mechanics, the theory, the explicit Newtonian mechanics.

So in a sense, this is a very good example to see the course of what Kohn calls the structure of scientific revolutions. In the classical version, collisions are described by giving the velocities of each particle before and after the collision. Think of like Brownian motions where particles of a liquid or gas hit each other. So you have a state of velocities for such particles before colliding with one another

and after the collision itself. So essentially this whole collision mechanic is based on an initial condition before the collision and a consequence, the state and velocity of particles after hitting one another. Nothing is said about the phenomenal collision itself. This is treated as sort of black box, whose inputs are the velocities of all particles before collision, and its outputs are their velocities afterwards.

The actual paths of the particles are irrelevant to this theory. So again, you see, even in terms of theory we don't always have everything explicitly defined the very term collision in classical cart collision mechanics is itself something of a vague term it is treated as a black box. All we need to know about a black box, namely a system of colliding particles,

is that what goes in it and what comes out of it. Obviously through this observation, kinds of invariances of behaviors of particles can be derived and that's all we are interested in therefore all basic notions we need to describe classical collision are finite non-empty set of colliding particles let me uh one second let before sharing this screen as we have some formulas here let me just get a cigarette sorry i i think i left my cigarette downstairs one second one second

can you see that share the screen yeah we can see it okay therefore all basic notions we need to describe classical collisions are finite, non-empty set of colliding particles P, a set of two time instances T, T1 and T2, where T1 denotes a moment before particles colliding and T2 denotes a moment after the collision, and a velocity function V assigning a three-component vector, one component for each

direction in the space, to each particle at each time. Further, in order to formulate the theory's fundamental law, we have to introduce a real valued function called the mass n of each particle. It's essentially a function, which is constant factor over time. We shall come back to speak on the differences of semantic and functional status between velocity and mass later on. Now the only difference between velocity and mass we can notice now is summarized in their respective formal characterizations. While velocity is time dependent, vectorial function whose range are triples of real numbers,

mass is a time independent scalar function whose range are the positive real numbers from what has been just said we can already figure out what the base sets of the models will be we need three base sets the set p of particle the set of incense t and the auxiliary base set r of real numbers that's all we need so this is you see when we talk about uh set theoretic modeling. This is essentially, you don't need to think about some sort of high-end set theoretic account. It's just we need to couch the kind of elements that we work are necessary for

our theory in terms of set theoretical relationships and memberhood. Nothing else. Reza, I have a question. Why T stands out here? Like, I mean, it's because... Well, key is what you might call to be essentially an implicit factor for us to behave, sorry, for us to observe the behavior of our black box. You remember that I said that the collision itself in the classical collision mechanics is not defined. So this term is under-defined, which means it's not defined.

We see collision as a system, as a black box. So T is what you might call to be essentially a variable that allows us to observe the behavior of what goes into the black box and what comes out of it, namely the status, namely the velocity particles before collision and after collision t1 and how come and how composition is not here well position is here but the thing is that in classical particle mechanics position isn't I mean we are

talking about pre-Newtonian as I mentioned this is essentially a pre-Newtonian theory. In Newtonian and also later on in to Gibbs and Boltzmann position will be added and position of course will be non-theoretical and non-theoretical factor or element but right now we don't have position. We are literally in the realm of Newtonian mechanics. We just want to see how the logical structure of this pre-Newtonian collision mechanics works, make what is implicit

in this pre-Newtonian mechanic explicit and then show that in fact it corroborates Newtonian mechanics, particularly the second law. so MP you remember MP was the sounding for what partial models okay the potential model Potential model, yes, or possible model. I mean, translations are different, but yes, potential model. So potential model of CCM, classic collision mechanics.

X is a potential classical collection mechanics, where X is a member of NPCCM, if and only if there exists P, T, V, and M, particle, time, velocity, and mass function, such that one, X is a tuple and well-ordered set. As I mentioned, when we are talking about well-ordered sets, all we are interested in are not just the elements that belong to the set, but also the explicit relations between such elements within the set.

So this is why we use instead of brackets, irregular brackets, these kinds of brackets, well-ordered brackets set. P is a finite, non-empty set. Why? Because we are talking about particles, obviously. So we need to have at least one particle. T contains exactly two elements, T1 and T2, time before and after collision.

region, V is a product of particle and time mapped onto a spatial framework that can be couched in terms of real numbers. We talked about like R3, X, Y, and Z axis. m, mass, maps particle to real number and for all p, a small p, belongs to the set of p or particles, mp, m of p or mass of the particle is greater than zero.

Now in order to obtain actual models of CCM, we just add the fundamental law of this theory to the conditions above. This law is the so-called law of conservation of momentum, which says that the total sum of the products of mass and velocity of each particle must remain the same before and after the collision. So in that case, mccm where x such that x is a classical collision mechanics x belongs to m of ccm if and only if there exists pt vm such that x is the tube called pt real number velocity and mass

x belongs to partial model of CCN, and 3, the sum of m of p v of p t1 equals to the sum of p sorry equals to sum of m of p v of p t2 where basically for both sums p belongs to the set of particles so you can i mean this i don't need to probably make the render this formula intuitive it's just simply saying that the mass function for each

particle the sum of mass functions for all for each particle before time before the collision and after the collision for such particles is equal Is it clear so far? I'm not hearing you. It's rather clear to me. Yeah. Okay. Oops. One second.

Now, can you see it again? Yeah. Now, in order to construct relativistic collision mechanics, which I'm not going to discuss it because it's more complex, we made two modifications to the previous frame. First to account for relativistic effects on moving bodies mass must be taken not as a constant function of each particle but as a velocity dependent and therefore ultimately time dependent function that is a typification of mass that is the typification of mass will change secondly we shall introduce a new basic notion into the theory i.e an existence function which

tells whether or not a given particle exists at time Ti. This would not make sense in classical physics, where particles are supposed to exist before as well as after the collision, but it makes sense when we come to the application of collision mechanics to elementary particle physics, where some particles appear or disappear in the course of collision. There is nothing is specifically relativistic about this existence function. In principle, it could also be introduced in the framework of classical collision mechanics. But since this introduction was prompted by the experimental results in high energy physics, where the only work of framework is that of relativity, it seems natural to include the existence function only in the relativistic version of

of collision mechanics now let us denote existence function by e small e a small is defined on particles and instance we agree that if e takes the value zero for a given particle p at a time this shall mean that p does not exist at that time if p exists at a given time then e takes the value one so essentially we are talking in a thinking about a very computational framework and literally you can say that Boltzmann's revision of collision classical collision classical collision mechanics from this

This perspective of introducing the existing function led to what we today understand as information theory, but also the science of computation. The introduction of E as a new primitive leads to the addition of a further auxiliary base set to those of CCM with the set 0 and 1, which is of course a subset of a natural number. So having reconstructed the basic structures of collision mechanics, we want to proceed to more important, more complex theory which also serves as a paradigm for

many discussions in the philosophy of science. Classical particle mechanics, CPM, is profitable to reconstruct CPM immediately after collision mechanics because CPM can be regarded logically as well as historically as a generalization and an enrichment of collision mechanics. The enrichment mainly consists in adding the concept of force to the concepts of collision mechanics. This allows us to treat a great number of dynamic systems which collision mechanics in its original form was not able to treat because its range of applications was much narrower from the beginning so you see

essentially the difference between CCM and CPM in this regard is not the theoretical core this the theoretical core we can say that the logical structure of both are commensurate with one another they are almost identical it's just that we add new factors to CCM And what comes out of it is sepian. Now sepian in that sense, what you might call to be the consequence of expanding the range

of intended applications without fundamentally changing the core of CCM, the core of our initial theory. CPM deals with all possible motions of particles which are considered as mass points in a space and it is assumed that these motions are caused by forces. The concept of force is not explained any further but introduced as an undefined basic concept. In any case, force must always be a vector value function. Just as collision mechanics historically is connected with the name of Descartes, so CPM

is associated with the name of Newton. Therefore sometimes it's called Newtonian mechanics. But this name is a bit misleading because modern formulations differ from original Newtonian formulation in some essential respects. They basically are generalization of the original formulation, but more importantly, but more important is the fact that CPM is the only mechanical theory attributable to Newton. Among others, we also developed hydrodynamics and mechanics of rigid bodies. So the expression Newtonian mechanics is ambiguous. Now in the following definition, sp denotes the function obtained from s by keeping argument

p of s fixed. sp of t equals to sp and t for all t belongs to capital T time. C prime 1 denotes the inverse function of c1. existence is guaranteed because of the requirement that C1 to be bijective. So, in this sense, partial model of Cpn, X is a potential classical particle mechanics. X belongs to mCpn, if and only if there exists pts a small s m f c1 c2 such that x is such a tuple i don't want to read it

it's too long pts are non-empty sets c1 maps time to real number and also c1 maps t to three-dimensional space, they are both bijective, meaning that you can go back and forth. S is a product of particle and time mapped to S. The composition of C2 and SP and C'1 is a smooth, namely differentiable for all p, a small p belongs to capital P,

and maps p to three-dimensional space denoted by real numbers, and function f is a product of a small p, t, natural numbers mapped onto, again, space denoted by real numbers here again p of time and s are set of points of a space so this is again in response to spitliana mentioned position now we are we are in the business of the position of the particles. Essentially our particles are only defined by their

time and their positional space, points of space. Instead of considering just a discrete set of instance as in CCM, we now take something like a continuous time interval, capital T, during which the particles are considered. C1 and C2 are coordinate coordinations of time and space respectively and separately which is an essential feature of classical theories. Capitals, sorry, C1, this should not be actually capital C, it should be just a small c. C1 maps points of time into real numbers while C2 maps points of space into real three vector. Cx,

Again, sorry, I'm sorry for these. C1 and C2 are bijective and therefore induce the structure of R and R3 on T and S, time and space. In this way, time and space are supplied with their usual classical structures. A small s is the position function which assigns to each particle p and point of time t a point of a space, namely that point of a space at which particle p is situated at time t. p's position at t, literally speaking. So, S of P and T equals, sorry this shouldn't be A, it should be alpha, means particle P at time T is at position alpha.

Usual formulations of mechanics introduce a position function which is differentiable, at least twice, with respect to time. Since T and S, time and space, have no proper intrinsic structure, differentiability of S, a small s, cannot be formulated without using the coordinations C1 and C2. We cannot require that a small s itself is differentiable. This makes no sense. but we can require that S2 composition, SP composition, S prime 1 be differentiable. If we look at the following diagram, we see that this function goes from R to R3 and therefore is suitable candidate to be differentiable. SP might be called the position

function of particle p or p's path m as in ccm is still the mass function i will i will show you the diagram uh but wait for for a moment we don't we do not want to identify a space and time with r to r3 which conceive of as purely mathematical entities points of space and time are not mathematical entities they are physical entities in order to take account of this fact we have to introduce the base sets capital t and s time and space this is in accordance with the genuine newtonian spirit in classical mechanics which views the space and time as real or absolute you remember Newton challenged to count.

Accordingly the position function small s, other than say the velocity function in CCM for which no such a spatial identification is needed, has to be viewed as assigning physical real places to particles at given times. Also a So a rather rich structure of space and time is needed, namely the structure given by the corresponding theories of space and time. These theories, however, in classical mechanics are not treated as explicit parts of this theory. They are presupposed implicitly, so it does not seem adequate to make explicit in a reconstruction of classical mechanics all the structure and axioms for space and time which are used implicitly. In these circumstances, the simplest way to get the desired structures is to put them

on time and space from outside, namely via C1 and C2. If we want to reconstruct theories of space and time, we can put them into a potential classical particle mechanics without difficulty. So essentially, now you see that the difference between the model and M and MP is really the issue of explicitization. There are some information that literally are encapsulated in our model implicitly. You know, they are containing our model. We do not want to touch them at that point. If we are going to think about a potential model of classical particle mechanics, then

that's when we open this capsule of implicit assumptions here with regard to the space and time. And then that's when we move from M to MP, model to potential model. We just have to take T and S as base sets of corresponding theories of space and time, carry along with them other basic notions. C1 and C2 then can be abundant and makes sense to require differentiability for a small s directly. Actually, in a strict treatment of CPM, we should say that C1 and C2 do not really belong to the conceptual framework of the theory, or perhaps more exactly, we should say that T, C1, and S, C2 respectively should

be treated as single concepts. This is why we treat second part of potential model of CPM for in the previous series of formulas that I mentioned as a characterizations of S as well for reasons of expository elegance and perspicuity however we have differentiated between T and C1 on the one hand and between S and C2 on the other. Now if we say that function f, a small f, called force has its arguments, not only particles and instants, but also natural numbers, each of these natural numbers, i, is intended to indicate a particular kind of force in the following way.

So when we are looking at classical collision mechanics, where we have actually positions and force, then we need to have different arguments for what we call force. what you might call the variations of our f, a small f, function of force. And I, through this diagram, sorry, I will show that why we need to have different arguments

for force. The fixed I as a third argument of F, obtaining FI being the product of particle and time mapped onto vectorial space. FI is called the Ith kind of force, especially if also fixed, a distinct particle P, we obtain fpi such that t is mapped onto, again, vectorial space, which is called ith component force acting on p. Thus, f of pti equals alpha means the ith component of force acting on particle p

at time t is alpha. In this way, the third argument of f of f are used to generate a great variety of different kinds of forces at Y, which for the sake of elegance, and again, perspicuity, comprehended into one single function, F. Take the following example. A particle P1 suspended through an elastic coil from a solid support, and the whole device is falling towards the air surface. In an idealization, which is typical for particle mechanics, the coil is imagined to be weightless.

And the support and the Earth are viewed as two particles, P2 and P3, respectively. Moreover, the support and P1 are both positively electrically charged so that they repel each other. Finally, we take into account the fact that the device is falling not in vacuum, but within the Earth's atmosphere, so that there is air resistance. It is clear that on such an analysis, there is not just one force acting on P1, but many. We could list them, which means we could use natural numbers to refer to them. For example, F of P1T1, force 1, could be the pulling force of the support on P1 at a given moment T.

Fp1T2, the force of gravity, the Earth's attraction on 1. FP1T3, the electrostatic force of repulsion, and finally FP1T4 would be the air resistance to P1's motion. On this assumption, the dynamical scheme of the system referred to P1 would be like this. Now, by indexing particular forces by means of natural numbers, we can consider as many forces as we need for analysis.

And essentially, when we are talking about many forces, are we talking about M or MP? So I want you to understand the difference between M and MP at this point. I can't see the sidebar, so you might as well. Are you still there? Still here. the question question is the difference between m and mp yes so when we are talking about you see as i mentioned that we don't need many definitions of force

okay we have only one argument we have force and that of course the note the concept of force, the function f, can have many form of arguments. Now when we are moving from just one function f to many valued functions, f, are we talking about mp or m? I think we still remain in the field of MP. Yes, we are actually, we are moving from M to MP, yes. Precisely because this is explicitation.

Essentially, whatever we do explicitize about the arguments of our concepts and functions, we are in the domain of MP at that point. Can I ask a more general question about how the movement is happening in it? It's an attempt to track the movement from one theory to the next as… We are essentially in the same theory. Right. The core of our theory is preserved. The core of our theory is preserved. It's about extracting explicit analog… basically simply we are in the business of unpacking the capsulated core

concepts so would this way way of describing be able to go about what Kuhn calls revolutionary science or is this no we are still in the course of the normal science right then you see the thing is that once our M is changed fundamentally or not even fundamentally let's say moving from classical to relativistic collision mechanics you see in the classical collision mechanics we thought we took it for granted that the particles that

collide with one another after the collision they will survive and we can study them right so essentially we okay think about it like this we have a black box in the classical collision paradigm we see that inputs go in particles go in and they come out and of of course the black box is the collision itself and they come out with various kinds of behaviors it was on the presumption or the presupposition that in fact something comes out of the black box

now you see even though relativistic collision mechanics It still works with the same kind of constants. For it, such presupposition no longer holds. Particles after collision might disappear, actually. And of course, once we presuppose that by necessity particles that have collided should survive

it so sorry sorry this uh okay by virtue that we no longer presuppose particles that have collided should survive in a relativistic framework we can now have a different kind of mp and range of intended applications. Fundamentally different. Could this be seen as a kind of deep challenge to the notion that

science actually undergoes fundamental revolutions? Yes, I don't think that there is such a thing as a fundamental revolution. science. And I would say that even more so, there is no such a thing as would Selaar's, I mentioned this in the first session, Selaar's complete scientific image. I think these things are a little bit of anthropomorphization of the course of science. Science, the cognitive advancement and evolution of it can be preserved without resorting to something like either a fundamental change or revolution or convergence upon a complete scientific image.

So allow me to resume. I know that you guys have been a little bit having brain nailed down right now. I just have a question. The material you're reading from is in, it's your own material that you've written, right? Yes. So basically, you know, this is actually, it is me trying to formulate some of the stuff that a Stegmuller and a Sneed have already been talked about. The material, in fact, when you read the Stigmular, is already there. I'm just trying to clarify some of these aspects. But basically, believe me, these are the stuff that no philosopher other than Esnit himself can go through them.

I just try to crystallize them to kind of give a kind of exemplification of what's going on. Can I ask a question of if you were willing to post them to the classroom? Sure. Okay, that would be super helpful. It's been about three years since I did physics in high school, and I got good grades, but it's back in the past. I always sucked at physics. I just hated physics. No, no insult to this, Milana. She won't like this comment. But literally, when I was at high school, my physics grades were like the worst.

It's only when I actually captured what mathematics applied math is trying to do that I got good grades in physics. But literally, I mean, you see, physics, to be honest with you, is far more difficult than mathematics. It's messy. my god I should record this I should record play to some of my friends literally I mean I remember my instructors first year of my university and school they were coming they were physicists and they were trying to you know teach us system dynamic analysis and they were bragging about

about how physics is the ultimate thing. And then next day the mathematicians, the mathematical professors coming and say that, oh, don't listen to these people. They're trying to pervert you. Mathematics is the real deal. really after all these years now I have come to this conclusion that being a physicist is like being a philosopher. You have to deal with so much of messy problems that are nonexistent in the domain of mathematics. That's a nasty subject to be sure.

Nasty subject. All science is a nasty subject, so has philosophy. Never coming into nasty subjects. Not for too long, but I just remember like I was like, you know, when I did it, I was did because it was the first year that we started doing calculus and physics and so what i wanted to do is we were doing this experiment where we're like dropping things and you could like use some equations like approximate air resistance but like i want to go all the way but then like the formula for like the whole like taking into consideration air resistance uses like hyperbolic trigonometry. No, absolutely.

I mean, literally, I mean, I'm sure as we'll now confirm, literally, when you wrap, even if you do rudimentary mechanical Newtonian mechanics, the range of parameters that you have to work with and you have to account for, it's just mind boggling. sometimes I think that how these physicists do actually think you know you just literally cannot I mean how what kind of cognitive resource do you have it's just not possible this is very flattering okay let's let's get back to our uh parochial collision classic classical collision mechanics

and uh post it to the the google classroom for the parochial minds Okay, one second. By the way, I found a very, very good essay, really detailing, it's not easy though, very, very detailing all of the relations between MP and MPP core, expanded core, various kinds of reducibility relations that can be obtained from them. So as being capable of thinking, Kony and Prada of normal science and scientific revolution,

I will definitely post it. Please, someone reminds me so I can post it on the Google classroom. What would be the sort of signifier for like an appropriate effective reminder to you i don't know i'm sure that you can find a better trigger warning for me than anyone else okay let's start Sorry, I have a problem to share this screen.

It just doesn't, okay, here. Can you see it now? Yep. By indexing particular forces by means of natural numbers, we can consider as many forces as we need for our analysis. Once we have determined all the forces acting on a particle P, for some purposes we might want to know what the total force acting on P is. That is to say, the so-called resultant force on P. So we might have different arguments of function F force, But essentially what we need to know about

in this theoretical framework is really simply the resultant force. This is given, i.e. defined by the expression sum of F of P, T, I, I is a member of natural numbers. This is a vectorial sum since the single F of P, T, I are vectors. We apply the rules of vector addition and make use of the convention that if there are only n forces, then F of P, T, J equals zero.

Actually this should be i, my apologies for, these are all, this should be i, and also for every i greater than n. From a purely formal point of view, sum of fpti, where i belongs to n, is the sum of an infinite number of terms. Nothing precludes the possibility that in some applications of mechanics, we should like to decompose a given dynamical system in an infinite number of forces. Well, with sums of infinite terms, we have to be a bit more careful than with sums of a finite number of terms. Or we could get an infinite divergent series

of force values with no definite total sum. sum and in fact those of you have a little bit familiarity with the analysis of dynamic systems this usually is the bug it's something that constantly crops up in the analysis of dynamic system precisely because dynamic systems have multiple trajectories according to how you frame the forces, the equations for forces. We do not require, and here we differ from other treatments, that the total sum always be convergent. We do not need to require this because in actual models,

this logically follows from Newton's second law. In order to give a legible formulation of Newton's second law and thus of the model of classical particle mechanics we introduce some notations. AUx of Cpn if X is such a two-pole then one for P members of capital P, Rp, R being mapped to R3 is defined by Rp equals to composition of s2 sp c prime 1 2 r is a product of p and r mapped one and two

r3 is defined by r of p and alpha equals to rp of alpha now the model can be defined as such. I don't read it but you can see this. So essentially the third term for all p member of p and alpha member of r mp of p d2 r p r of p and alpha equals to sum of forces

applied to particle invent inverse position inverse function position applied to particle alpha and I, I belongs to natural number. The third term is essentially what you might call to be the encapsulation of the second law of Newton. The core of our theory. Obviously, in exposition of these kinds of uh logical structure i have had to you know get rid of so many details to follow a step by a step

how the logical structuration logical logical reconstruction of your theory actually make explicitly what your theory is actually and of course our theory we started from pre-newtonian classical collision and then we see that once we define terms explicitly then we will end up from the perspective of logical reconstruction with what was already the case Newton's second law the core of the theory

I feel that you might have been a little bit taken aback to put it mildly by this formalism. So let's not go to Karna because Karna is the same words. So let's just hear your heckling suggestions, so on and so forth. I mean, just simply put my case, it's late. I feel like with some previous exposure to the specific arguments, like in the context of collision connects or whatnot,

I may have been able to follow a little bit, But without that, I really have to try really hard. But that's like about it. Now, Christian is someone who tells me that, you know, he's putting all of his focus these days on mathematics, particularly the category theory and stuff. And sometimes I tell him that, you know, many people today, if you look at Twitter, There are, oh, topos theory this, category theory that, proof theory this. Well, literally, now you can see that these kinds of mathematical stuff cannot be just applied whimsically.

When you actually put one of your foot in the realm of formal mathematics and the other foot in the reality of physics, then you are in a whole new game and this is not really elegant from the perspective of what you could do in the realm of pure mathematics no I mean I do this out of like purely just practical like it's what I get paid for yes yes but I do it you know the more that I realize you know that like as As much as for practical purposes I've made concessions away from philosophy, the more

that I realize, try to actually, in the most robust manner, tackle the problems that I'm trying to solve, I will just, there's no compromises. I have to return to philosophy. There's no... Absolutely. I think that philosophers usually think that mathematics is closer to philosophy. I would say that physics is far more closer to philosophy than mathematics could ever be. And I think that any person who glorifies mathematics as a kind of deep ontological discipline or something that is essentially in a spirit close to philosophy,

simply indulge in romanticism. I saw this funny thing. This is really just an aside, but it's pertinent to what you're saying. I don't know. I saw someone share. It was a philosopher. It's good that they shared this, but it's just on Facebook. Let's hear Espitlana if she has some comments on this. Knowing that she's coming from the physics background. What was said was it was like John Bize was like, you know, propping up one of the people that was close to him. He's saying category theory is like the logical theory of logical theories of logical theories. What the fuck does that even mean? Like that has no kind of, there's no philosophical profundity in any of that.

Like, there's nothing to that. So all these people are completely impoverished in all the philosophical argumentation around all of this. Esvidlana, do you want to make observation about this relation between math and physics? for me I'm not someone who would like prioritize one field over another for me it's like I know mathematicians and physicists and it's like people having different work to do you know different things at stakes

and different questions are asked completely different. Maybe in case of theoretical physics, it's less clear because they do quite a lot of maths there as well. Like for me this, You know, physicists, like the kind of thing I learned is that math is a language of physics. And this is something like, I mean, I was brought up with this kind of understanding, and I still cannot really give it up. Yes, yes. so it's this it's not very straightforward relationship it's yes I mean to be honest

with you the more after the last session we had a very interesting conversation with you and I a few others I think the way that I see physics I mean if we were to use a philosophical metaphor for physics it's much more in tandem with Kant calls transcendental psychology or sorry transcendental logic transcendental logic it is not just about the understanding of logical rules it is about how these logical rules or in this case mathematical objects or a

structure mathematical structural entities can be applied non-arbitrary way to an objective round the physical universe of course to say something like that I still do insist against Kant that we should take the idea of mathematics very very seriously the more we can enrich the universe of mathematics precisely because it's the organ of a structure the more we have the possibility of making new objective claims about reality however However, Kant's dictum and cautionary tale should be taken very seriously.

Because even though we can go and talk about a structure or logical structure in its own terms as unbound without any regard to application, we should at some point constrain such rules to that of our experience of reality. And then at that point the real question as Kant would have said that would be how are we sure or how can we be sure that the roots or the structural entities that we have made

in the realm of pure logic and mathematics can be in a non-arbitrary way applied for experience of the world or physical universe that's that's i think the the real question that ultimately bugs me. Can I respond to that too? Sure. It's part of how I understand this problem of thinking of the object, or as Kant says, the thing in itself, because in some way it forces us to think of it as a constraint, But at the same time, we can't say that we have these constraints given to us through some empirical way, I suppose.

So it requires that we, but at the same time, in order to even talk coherently about structure at all, it seems that we in some way need to posit the object as constraint. Otherwise, I feel like we run the risk of thinking that structure is given. Yes, yes. which is the other side of the myth of the given. I don't think that a structure can ever be said to be given. You see the myth of the given, and those of you who might not be familiar with this, we have repeated this.

I mentioned this to you, that one of the greatest challenges that Wilfred Sellars, American philosopher, posed against empiricism was something called the myth of the categorical given. The myth of the categorical given is the idea that our knowledge, our knowledge, this is really important, our knowledge of the world at some point bottoms out in some fundamental correlation with reality itself. Like for example empiricist Humean sensedatum where the color red is red and is giving its

own structure to us so we know what that red primarily is. So the idea of the myth of the given ultimately is about knowledge. idea of a structure or attending to the dimension of the structure is not primarily about the knowledge of the world. It is simply about the structure and hence I don't think that we should say that the idea of logic as an organon, pure math, attending to the dimension of a structure without object is a variation of the myth of the given. Why? Because the myth of a given

is about knowledge. When we are in the domain of a structure, all we take to be serious is really the coherence of our logical structure. We do not even know, we do not even want to know about the knowledge of the world it's just at the second phase at the second phase that we say that these structures once applied to dimension of our experience can yield knowledge so in the primary phase namely prioritizing the structure over sense data or empiricism i don't think that

leads to the myth of the given because it's not about the knowledge of the womb it's simply the idea that all we do have is a structure anything other than that leads to what sellers call the myth of the given parochial empiricism sorry did you just say that to say structure is given is not the same thing as to say so the same thing as means of the category categorical given categorical given yes it is not because categorical given is about knowledge of the objective world a structure does not require

objective knowledge. You see in German idealism we have at least two terms. One, what Kant called understanding and the other one is called reason. We can even go further and saying that in a Hegelian sense understanding one and understanding two. What is the difference between understanding one and understanding two? Understanding in a special way and reason as general understanding. Well, the thing is that understanding always in the first

sense, understanding one, the special case, is about objective knowledge of the world. Whereas reason is not about understanding of an external reality. Reason is about construction of the infrastructure by which we can in fact come into contact with an external reality. So from this point, myth of the given can only be applied to understanding one, the parochial sense of understanding. But if we take understanding in terms of reason, in terms of constructability of rational

resources, namely logic, mathematics, so on and so forth, then the myth of the category of given no longer applies. Because it is not about objective work. It's simply about one thing, that every contact with everything always is regulated and constituted by a structure and not by sense by sense that one although empiricism naive empiricism what if we say that mathematics is not the structure of physics but the technique or the techniques for sazik's problems

Can you elaborate what you mean? Okay, so I can understand this question, but would you be able, Meredith, to elaborate on the point of how you set apart technet of mathematics, technology of mathematics, from what you might call to be... a structuration. Sure. So, if we think of mathematics as a technique, so I'm thinking, for example, you know, we develop calculus

to understand the movements of heavenly bodies. You know, we develop linear algebra to solve, you know, linear equations for dynamic systems. And rather than saying this is our toolbox by which we are solving problems about the physical world that physics asks. Whereby the structure is the whereby is the structure of physics a linear, like a matrix or an equation?

If mathematics is the language, then... I guess it's similar to in philosophy, you know, I have different techniques to discuss philosophical questions. I can talk about dialectics. I can talk about a Socratic dialogue. Or I can talk about phenomenology or different sorts of techniques to address philosophical questions. Is that the language of philosophy? But it's a good question. Something I have to think more about. but I wonder, you know, we've been talking about mathematics for so long as the language or physics.

Is this really the case? It is not the language of physics. I would say that only in a very specific sense of language, in the sense that Carnap thought about it. So Carnap following Frigge thought about language not as a representation of the world, you know, and from this perspective Frigge and Carnap are in one trajectory with regard to the notion or the concept of language. Szilard, Zewittgenstein, Kant are on the other.

In the sense that even though Sellars doesn't think that the concept of language should be considered as a representational function, as if you're representing an external reality, nevertheless he thinks that at some point at some level at some basic rudimentary level the function of language bottoms out in rudimentary representations to be honest with you I think that Carnap to me is on the right

track so as Freedia rather than Sellars, Wittgenstein and Kant. When we are talking about the language of physics we do not mean it in the sense of mathematics allows physics to arrive at the objective facts that it has precisely because mathematics already has some sort of objective representation of the world in the sense that the laws of thoughts correspond to the laws of being or the world. No, precisely because mathematics in this sense is a calculus also, is a language, is general language

in which all we do is diversify and elaborate the concept of a structure with the understanding the concept of a structure is not given in advance to us. by an external reality or the world or the physical universe. And from this perspective, yes, mathematics can be said to be the language of physics, but only in this very specific sense rather than saying that mathematics or the mathematical structures or the objects of mathematical or the or mathematical entities as objects have already some sort of correspondence to an

external reality with which physics simply elaborate physics is not a hand the maiden of you know mathematics it uses mathematics to its advantage in in order to navigate an objective world. I wonder what would mathematics use be beyond physics? Well, the same thing that I said about physics can be extended to every other discipline and their relation with mathematics, biology, philosophy, so on and so forth. in the sense that if we really try to see mathematics generously the mathematical

universe generously and by that i do not mean what traditional mathematicians understand mathematical universe i have much more in mind i basically think of the so-called computational territorialism there is a fundamental correspondence between logics mathematics and language and in this sense mathematics language or logic can be said to be the way of elaborating the concept of a structure the elaboration of the concept of a structure is in principle not necessarily i'm saying in principle in italic in theory can lead the

enrichment of our encounters with objective reality that's all i want to say with regard to the mathematics and that's why i think mathematics is important but many claims made by mathematicians either verge in verge on you know kind of metaphysical dogmas in the sense that like roger penrose said he thinks that there is some sort of correspondence basically that the universe corresponds to the mathematical that's just that's pure metaphysics this is something that you have learned in philosophy 101 to avoid or even more insidiously, to think of mathematical structure,

logical structure, linguistic structure, as having a one-to-one correspondence with the basic observational facts, in the sense of Carnap's logical empiricism. As Carnap showed in our Bible, mathematical structure does not in fact correspond to our atomic empirical facts. Atomic empirical statements are in fact caught up in what you might call to be a structure. And a structure is not just one entity or a mathematical object.

The notion of a structure is always like a web. It only is a structure in so far as it stands with NRE relations with other entities inside that logical or mathematical domain. And that's, I would say that is for me the premise of actually think about these problems coherently without either sidestepping into a kind of metaphysical understanding of mathematics or a romantic idea of mathematics, where all the structures that mathematics produces can be at whim applied to an objective rule.

Maybe Laura and Sepi, they have been very, very silent, so that's Joven and a few others, Eduardo. Maybe they can, I think it's time for them to talk. I guess I was wondering, I'm still like a little hung up on the potential models, partial potential models, and like since I also haven't taken a physics class since I was in high school, it's a little bit hard to visualize what's included in these different sets in

like abstract terms. Yes. I mean, we're using a concrete example, but since I'm just not like a physicist, I'm just like constantly have to go back to remember like what each symbol is standing for. So I'm not even, yeah, I'm not sure what level of like detail is necessary, is like important for understanding this distinction between the different like subsets of models. but um i have a solution how about uh i make uh i don't i will write an essay 3000 word essay i will just make a table where i can i can correlate for example m in this theory stands

for this in this theory it stands for that mp stands for this yeah yeah mp stands for this how about that that sounds good i think that yeah okay if it's not like very annoying i would be also curious if it's possible to kind of display how the same theory can be thought of in both the statement view and the non-statement view if it's possible to like take a particular theory and like um kind of extract its structure through these two different lenses just to have a bit of a more like concrete idea of sure sure absolutely if that's if that's not like super annoying to do i mean i've tried to read no no i apologize for not having done this before but this is a perfect

suggestion okay i will definitely do this absolutely yeah i've looked at some like secondary literature that tries to explain these distinctions and i've found that either they're incredibly cursory and don't go into any detail to really explain these differences yes or they or they become highly technical and I'm not really sure how to map that to real examples. Yes, okay, I will definitely do that. I think the best thing is that thinking of table, and please, everyone can contribute to this. we can have a table where we can differentiate, for example, the differences between M, MP and MPP, core, expanded core constraints.

We will, without going way too technical, we will look into what they mean for a specific theory, like classic collision particle mechanics. and then we can also think about what do they actually stand philosophically like for example we know that MPP stand for what a state builder calls observable facts and by observable facts I mentioned last session he doesn't mean like something like Carnapian but he means simply theory-laden statements. Okay, so that's, yeah, definitely. I think this is something that we

should definitely do together. I will make the basic table, put it on the classroom and anyone can contribute to that. Okay, cool. Thank you. Absolutely. Go on, Espitlana. I just had a very similar question. like i wanted to hear like in the case of for example when we have collision mechanics or even before like what would be the mpp is it like because i'm you are now forcing me to become a little bit even more towards you than i have been before the reason that i haven't talked about mpp

because things get a little bit even more messier than what I have talked about so I think that this whole idea of table is a good idea so we can have some sort of at least some semblance of correspondence of what it stands into what relation with other things yes yes no I I I try to weasel up out of the talking about the mpp but okay let's wait we're at our time too so i guess if i'm open to staying a little bit longer but just letting everyone know and class starts in like 20 minutes

right yes by the way uh i know some of you uh adam is not today i know adam laura you are also part of adam's class right yeah i am let me tell you that let me just promote my friend even though that's nasty but anyway we are in church i i to be honest with you i think adam is one of the greatest philosophers in our generation. I mean maybe my generation. The thing is that, the thing with Adam is that he's very, first of all let me introduce who this person is,

he's a winner of Beller Prize and if you know what Beller Prize is, you should know that he's one of the most fundamental prizes in analytic philosophy and science. So he wrote his thesis on entropy gradient via Boltzmann and he won a Beller prize. And literally for him the whole idea of continental analytic distinction is new and void. He just simply does not think about philosophy in such terms. In fact, he would be offended if he would say that, well, are you an elite philosopher or a continental philosopher, by the way?

He just doesn't believe in such things. And this is really the kind of philosopher that I would say that we need right now. People who do not pigeonhole the idea of philosophy in this or that kind of camp. And his ideas is literally his way of philosophizing is fundamentally profound. He's as much informed by complexity sciences, Turing, computation, Wittgenstein, As Deleuze, Lyotard, Jean Cattito, so on and so forth. I particularly Peter Adam you know I having been in contact with the work with

the art world mafia I call it a mafia and I hope he gets recorded I think that I I was burned by this whole notion of art. I just do not want to hear the word art in my class. But Adam is one of the very few people who actually convinced me that why art can actually be experimental in the spirit of philosophy. So he is really good if you are for precisely because of the questions that you pose at the end. I literally think that he can give you some arsenal, some slings and arrows down the line. Good to hear.

Which, I hate art world or I do like art. I guess I share your ambivalence. okay friends uh i think it's time for you to go have some fun it's a sunday you shouldn't just waste your time on the stupid philosophy go to the gardening have some good food and care take care of your loved one Thanks, Reza. Thanks, Reza. Bye. Thank you, everyone.

Goodbye. See you next week.