Maths & Ideas (Session 9)

Hello and welcome to the ninth session of Maths and Ideas with Reza Mankastani. I'm going to pass the mic off to him now. Thanks. Okay, so today we will continue talking a little bit more about the doctrine of configurations in URIM, and which is still we are trying to figure out the effects of analytic geometry on the conceptual basis of, you know, Enlightenment sciences that happened throughout the Middle Ages up to the

period of higher scholasticism. Also after that I will make another example and that would be Al-Khar Azmi's geometric solution to quadratic equations And these would be our two most important examples of how scholastic philosophers and mathematicians were capable of advancing mathematics via analytic geometry. So, before moving forward, any questions from the previous session?

Are you familiar with Nick Szabo's piece about vending machines being primitive smart contracts? Or like mechanical smart contracts in the sense of like self-enforcing a contract of sale? I haven't read that piece, but I'm familiar with his work. I really like his work. Likewise, it was occurring to me as I was like, after watching the part about mechanical implementation of prosthetics for geometric problems, and the sort of triad going on there, that he does like really a lot of the exact same thing between cryptographic math, monetary economics,

and then whether it's like mechanics or computer science or like contract law, depending on the implementation that he's talking about. That vending machine example is just perfect. I will definitely check this. Is it on his blog? Yeah, yeah. Because his website is now defunct. I haven't been able to access his website. Can you summarize the example? It's just that a vending machine mechanically self-enforces a contract of sale. Whenever I get the vending machine in the background, whenever I get a Pepsi out of there that costs a dollar, the mechanical function of the vending machine is to force the case that one Pepsi won't be traded for $1

as long as there is Pepsi in dollars. And so he says that that's like a mechanical implementation of a smart contract, which is a contract expressed as software that can then execute itself. Given inputs that trigger its logic, you know, like the strike price on a stock is reached and therefore I have to sell or I have to buy or whatever it is is accomplished automatically. and that's like where he's going with this with this notion and he seems to be following like this very similar I don't know your whole diagram of how engineering and geometry to go together really kind of I think especially if you see the one way equivalence relation that is get a geometric solution from the mechanical one if you see that

as like what he's going for in terms of converting contracts into software if that's what you want to do then you engage in this like reciprocal reinterpretation between like math and contract law drives the process of engineering and like also like analog to digital conversion kind of follows the same diagram I don't know I'm just going to see if any of that like fit your understanding of how that prosthetic process works. OK, any more questions? Before we started, you were sort of diagramming the,

I think you were diagramming motion in motion. Would you mind going over that really quickly? I will go over it. OK. Today's, yeah, I will basically cover this much more details. The thing is that, I mean, from a historical perspective, you see that pre-Socratics, I mean, we were talking about that, how contemporary mathematics is somehow resurgence of some of its earlier historical concerns. So you see that in pre-Socratics, bodies are expressed in terms of forces, Democritus, Heraclitus, so on and so forth.

Aristotle is the one who somehow manages to freeze this motion, and that's basically the base of Aristotelian metaphysics in the sense that mathematics should represent pure forms as divorced from the messiness of physical motions that can be expressed in terms of spatial properties. Now you see that precisely because scholastic philosophy is both conceptually tied to geometric analysis but also is methodologically being limited to analytic geometry.

And insofar as analytic geometry, you can think about it as continuous variations rather than discrete instantiations. And you see that scholastic philosophy and mathematics is predisposed to form a landscape of mathematics in which everything is expressed in terms of motions. And then you see that the ultimate summit of this scholastic preoccupation is in Descartes and Spinoza. But then this project is being abandoned throughout the development of modern mathematics, and

it's not being in any way resurrected until very late in 20th century, but in a completely different form. Okay, so as I mentioned, the difference between So, Orim's approach and the modern approach is not simply the question of methodology,

namely that Orim, all he had was analytic geometry, whereas mathematicians of 20th century and 19th century had differential calculus and limit functions. It is actually also a conceptual difference. And this conceptual difference, as I mentioned, can be encapsulated in the difference between modern graph modern mathematical graph and the configurations namely geometric diagrams of worry word configuration just to be clear it means multiple

figurations figurations mean as in in or even lexicon it means representations or geometric representations, diagrams of variations of quality in one subject. So configuration means that a body that has different qualities at one time, how can it be represented? You can think about simple motion on a projectile, motion I mentioned it from in an Aristotelian metaphysics represents a quality or for example a tepid body a body that is at the same time cold and hot so the difference between modern graph and

configurations or immune configurations is that the modern graph immediately seeks to focus all attention on the distance actually traveled by the moving body by reducing it to an output. Now such a quantity of time at such a velocity transmits such a quantity of length and the relation of the equation L equals to Vt that I mentioned is satisfied with making this bit of x-axis correlated with the other bits of y-axis like this. Can you see this?

Yep. So this is completely can be articulated in terms of input-output output relation, which of course can be mathematically modeled by way of limit of a function, the so-called epsilon-delta description of a function, which is due to Agustin-Louis Cauchy, then Bolzano, then Weistras. Now configurations, on the other hand, represent a transit of forms or qualities, this whole

idea of continuous transition of forms, in which the area of any geometric figure in Orym's configurations represent the envelopment of the units of differentiation or individuation. This whole idea of amplitude is coming from Orym. The area of a rectangle in Orymian diagrams are, for example, uniformly uniform motion or uniformly deformed motion represents how degrees, differential degrees, different degrees, only vary over time, but how they are grasped within one single envelope, the so-called

amplitude. So by representing this differentiation, individuation, and variation as an area, Orym shows that he is capable of grasping intensities and extensions by means of a single intuition without departing from a tradition that carefully distinguished from them. So he, another meaning of configuration in Auriem's lexicon means comparison. what we need to know is that how these degrees of differentiation between

qualities are related to extensible properties namely the spatial properties of a subject or body in one single intuition so in this sense Aurin establishes a physical mathematical correspondence between the extensible properties of a body, spatial properties, and the material properties, the idea of material corpora that I mentioned in the Abyssinus triptych.

So you get basically how qualities, how forms, how intensities can be expressed in terms of spatial properties and how these spatial extensible properties can be translated back into qualities. This is the whole conceptual edifice upon which, you might say, the entire philosophy of measurement is built. So, as I mentioned,

Uriam's most famous book is The Configurations. It's a book in which he started to make a leap from the previous scholastic study of ratios done by Merton College on the topic of mean speed and move toward the study of uniform acceleration.

Uniform acceleration during the scholastic time was conceptually one of the most significant discoveries from a metaphysical philosophical standpoint uniform acceleration or uniformly deformed motion is not simply uniform acceleration in modern terms because motion represents quality so uniformly deformed acceleration represents how qualities can be combined together in the sense that they can vary or be differentiated over time so you

can have the configuration of qualities within qualities like heat hot within cold the so-called form of a tepid body or a slowness within fastness quickness acceleration so on and so forth so the emphasis on the idea of uniform acceleration needs to be understood within the overarching metaphysical philosophical project of a scholastic philosophy and not simply in terms of

what you might call to be Newtonian mechanics or today's understanding of what really uniform acceleration is. It was simply a conceptual question, a conceptual question about the intelligibility of individuation of forms or qualities and how these forms or qualities that are so entangled and are varied between one another can be rendered intelligent, can be articulated by degrees of more or less via the extensible properties of a body in this spatial attribute attributes so the first part of configuration establishes

the tenets of the geometry of figuration doctrine and that's applies to doctrine of qualities namely to entities which are essentially permanent or enduring in time and related to the intricacies of the internal configurations of qualities Orin, in the course of elaborating the doctrine of internal configuration, suggests how the theory might explain numerous physical and psychological phenomena. The second part of configuration describes how the configuration doctrine can be fruitfully

applied to motion, namely to entities that are successive. Do not think about motion in terms of modern concept of motion. Motion for a scholastic philosopher represents successiveness of qualities, which means that it goes beyond the idea of today's motion. So after describing the external geometrical aspects of the doctrine, Orendan goes to a detailed analysis of how actual natures of motion, producing some kind of essential configuration,

may well account for certain sonic and musical effects. And he moves from this to some sort of cosmological musicological doctrine in terms of harmonics. And in the second part he concludes with a discussion of how these essential configurations of motion go far in explaining magical and psychological effects. So we see his broadly conceived doctrine of configurations giving him a physical basis to attack magic, in fact, in scholastic philosophy, simply by rendering magic susceptible to a

spatial property that can be articulated in terms of degrees of intelligibility. He is the first philosopher who sets a systematic attack on the whole edifice of magic. We need to understand this idea that every conceptual apparatus that emerged throughout the scholastic philosophy which later on some of them not all of them some of them led to the advent of Galilean revolution were coming from this

entanglement between the conceptual resources that the study of magic gave natural philosophers and the logical geometrical resources that scholastic philosophers had so it was not just there wasn't such a thing as science or magic it was basically magic was a conceptual basis under which science was being studied and philosophy was really the system that was studying the logical connections of these concepts that were basically covered inside natural magic precursor to natural

philosophy. Simply by showing that all magical properties and all magical concepts can be articulated in terms of the logical connections and being reduced to the logical connections and ultimately to a spatial properties he has started to fundamentally make a rift between magic and science to the point that science became separated from natural magic which was the most dominant field of science in his scholastic time.

So as I said in the first chapters of configurations, Oren broadly develops the doctrine of configurations that gives him a physical basis to attack natural magic. Just as his elaborate discussion of the ratios of ratios give him a mathematical base to attack astrology. Finally, in the third part of configurations, Aurim returns again to the external geometrical figures used to represent qualities of motion. shows how in the comparison of the areas of these figures we have a basis for the comparison of different qualities and motions so we are capable of articulating

how intensities are being related to extensities in terms of areas of geometric figures and in so far as areas of geometric figures can be compared with one another in the fashion of Euclid's elements then we are capable of comparing different qualities and motions residing within subjects in the extensible bodies.

So there are two keys for proper understanding of Aurim's configurations. The first is that Aurim uses the term configuratio with two distinguishable but related meanings. One, a primitive meaning, and the other, a derived meaning. In its initial primitive meaning, it refers to the fictional and imaginative use of geometrical figures to represent the graph, to represent or graph intensities in qualities and velocities in motions.

therefore the baseline of such figures is the subject when we are talking about linear qualities or time when we are talking about velocities and the perpendiculars the lines that we were raising above the subject that is being extended in its attributes or extended in time represent intensities of quality from point to point in the subject or represent the velocity from an instance of one instance of time to another instance so

So depending on are we talking about qualities or we are talking about velocities, the horizontal lines and the second line, the perpendiculars or so gonads, represents either extensible properties of the subject or simply subject instantiation of subject from one moment to another moment and perpendiculars respectively represents either their variation of intensities or variations of motion philosophies

The whole figure, the horizontal comprising of the horizontal and vertical, represents the whole distribution of intensities in the quality, namely the quantity of the quality, or in the case of motion, the so-called total velocity, dimensionally equivalent to the total space traversed in the given time.

As I mentioned, the difference between these configurational diagrams and the modern graph is that they encapsulate the conceptual core of what it means to render something a property of substance or its potential or the potential of matter intelligible in one single grasp how you can connect intensities to extensities go back and forth decompressing intensities to extensities of compressing extensities back to intensities.

Equality of uniform intensity is therefore represented by a rectangle, which is its configuration. A quality of uniformly non-uniform intensity starting from zero intensity is represented as to its configuration by a right triangle. Similarly, motions of uniform velocity and uniform acceleration are represented respectively by a rectangle, a parallelogram, and a right triangle.

Now between a rectangle and a right triangle there are infinite amount of variations. How does this work from a Euclidean perspective? That leaves them positive light. Let me turn on the iPad. Okay.

Can you see the whiteboard? Oh yes, now we can. So you see the relation between uniformly uniform and uniform and uniformly deformed can be expressed in terms of a rectangle and a right triangle. We talked about this idea that the horizontal line represents a subject, either in its variations

of extensible properties if you are talking about qualities or in terms of how it is extended in time if you are talking about velocity of motion now regardless of that if the horizontal line in Orem's configuration always represents an object a body namely an extensible thing the perpendicular lines represents intensities now the thing is that we talked about at the end of the

previous session that these intensities if we are going to understand how they are varied in time or in terms of extensible properties we need to connect these perpendicular lines namely latitude of forms by a line which is called the line of summits now the line of summits creates these areas, as you notice, that allow us to articulate how intensities, namely latitudes

of forms, L of forms, are connected to extensible properties or variations in time, namely longitudes forms which can be quantified so by creating the line of summit we creating area by creating area we are capable of articulating the relation between qualities and quantities intensities and extensities the line of another name

for the line of form is altitude. That's in configuration or in goes back and forth between these two phrases when he talks about these lines. They can be also surfaces. Line of summit or altitude. Now imagine like this. So this is your subject, horizontal line, always a subject, extensible subject, a quantifiable thing. And the line of summit can be represented in different ways. Parallel. We know that the parallel is a diagram

representation of uniformly uniform motion or uniformly uniform qualities now imagine if the line of summits according to the doctrine of parallelism The line of summit, the opposite of it, that the line of summit will be perpendicular orthogonal to your line of subject.

From a Euclidean perspective, between the parallelism of the first and the orthogonality of the second, there are infinite variations of how the line of summit can be posited. The perpendicular is the remission of quality back to pure intensity. And the parallelism one, which represents the uniformly uniform motion, uniformly uniform

variation of qualities represents an equivalence to pure extensionality so between pure extensionality and pure intentionality we have an entire array of unfoldings of how intensities can be articulated in terms of extensities within remission of forms or qualities and their extensible manifestation.

Okay. So, now we see that for ORIM differences in configuration taken in its primitive meaning reflect in a useful and suitable fashion internal differences in the subject. Therefore, we can say that the external configuration represents some kind of internal arrangement of intensities, which we can call its essential internal configurations.

How qualities are bunched up together in one single subject. Again, like tepidness, like acceleration. cold fast and slow so on and so forth qualities always of the same kind so that can be expressed in terms of more and less in this way we arrive at the second and usage of a term configurations, the derivative one. Configuration in the second sense, abundance, the purely spatial or geometrical meaning,

since one of the variables involved, namely intensity, is not essentially spatial. Although O'Reilly tells us variations in intensity can be represented by variations in the length of straight lines. Waurim suggests at great length how differences in internal configuration may explain many physical and even psychological phenomena, otherwise not simply explicable on the basis of the primary element that they make up a body. Therefore, two bodies might have the same amount of primary elements in them, and even in the same intensity but the configuration of their intensity may well differ, therefore

producing the different effects in natural actions. So you see that from simply an analytic representation, geometric analytic representation, he's capable of articulating fundamental mathematical physical correlations about different bodies. The second key to the understanding of the configuration doctrine is that which we can call the suitability doctrine. It simply concerns with the nature of the configuration in their primitive meaning of external figures.

Very briefly, it holds that any figure or configuration is suitable or fitting to describe a quality when its altitudes on any two points of its base or subject line are in the same ratio as the intensities of the quality at those points in the subject. Now I will get back to this shortly. So every measurable... This is a quote from Orym himself. Every measurable thing except numbers is imagined in the manner of a continuous quantity.

Therefore, for demensuration of such a thing, it is necessary that points, lines, and surfaces, or their properties, be imagined, for in them, i.e. the geometrical entities, as the philosopher has it, measure or ratio is initially found, while in other things it is recognized by similarity. analogy of different areas or different geometric configurations. Although indivisible points or lines are non-existent, still it is necessary to feign them mathematically

for the measures of things and for understanding of their ratios. For every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of a space or subject of the intensible thing . For whatever ratio is found to exist between intensity and intensity in relating intensities of the same kind, a similar ratio is found to exist between line and line, and vice versa. For just as one line is commensurable to another line and incommensurable to still another,

so similarly in regard to intensity, certain ones are mutually commensurable and others incommensurable in any way because of their property of continuity. Therefore, the measure of intensities can be fittingly imagined as the measure of lines, since an intensity could be imagined as being infinitely decreased or infinitely increased in the same way as a line. This whole idea that you see it's only with development of modern, you know, post-17th

century physics and mathematics that mathematicians and scientists are capable of understanding body no longer as some sort of passive thing, but bodies always being endowed with force that can be expressed in terms of vectors. This whole idea that Aurim is capable of understanding a subject in terms of the perpendiculars, namely its intensities, being orthogonally connected to the subject line, is already a precursor to the modern understanding of a body as something always being endowed with

a vectorial force. So from this perspective you can see that, at least conceptually speaking, the idea of configurations, namely comparison of areas of geometric representations, is more connected not to the idea of modern graph, but to the idea of bundles, how to integrate different vectorial spaces. So you see that, for example, something like, you know, in today's study of geodesic curves,

are talking about a body that changes its vector something like this as it moves along a curve now the vectorial modern analysis of vectorial spaces is is about bundling up these vectors, how to integrate these vectors. With these vectors being, varying over time, or the body as it moves along the geodesic curve. The same thing is basically,

you can see is conceptually articulated by the doctrine of configurations. Questions? When you say bundle, that's like the formal mathematical structure of a bundle, like a set of vectorial spaces, like the integration of a set of vectorial spaces? Yeah, not in the sense of really like a bundle space, but bundle in terms of just algebra, Geometric algebraic geometric structures that you can express and anything in terms of one of the vectorial spaces Okay, and so like what's being done here when we can look at it is

like maximum compactness or a fit or compactness or efficiency of mappings of dimensions of a physical interaction onto the dimensions available to you discernible in drawing lines on a page right like up to like all the extensive properties available on the page, like area, width, height, all of that, and then plus one intensive property, which is slope or angle or whatever. Is that dimensionality, n plus one, one additional intensive dimension, which is jerk here and allows him to represent acceleration on a page? Is that like rigorously Deleuze's line of flight? Like that, like the... Yeah, yeah. Yeah, I mean, if you, if, yeah, I mean, basically Deleuze is quite explicit that he gets some

of these ideas from Merton School and OREM. Awesome. There is, there is in fact a footnote to OREM's configurations in ATP and a very, very extensive paragraph about OREM's in difference and repetition. So yes, absolutely. Excellent. Excellent. And does he ever directly reference Avicina? I was looking for any connections between them just loosely on Google, and I was shocked that, given what you said about Avicina then sort of developing this theory of Intensives' quality and quantity, that Deleuze goes to Spinoza rather than Avicina. Yes, because I think Espinosa obviously encapsulates and completely refines its scholastic concepts,

whereas in Avicina it's still in a different form, a kind of a scholastic tradition. Also, another thing that you need to understand, and this comes back to the idea that we were talking about invention or construction of a spatial invariance you see that the whole idea of intelligibility namely creating a point of liaison between intensities and extensities is being articulated here again it's the idea of unfolding a dimension the whole task is not simply connecting as I mentioned in

terms of like modern graph a bit of y-axis a bit of x-axis in terms of input output relation but it's really about of how you open the dimension between the vertical axis and the horizontal axis depending on what kind of dimension you have opened and how it can be geometrically modeled, you are capable of articulating new relations between intensible and extensible properties, qualities and quantities. We see that Orems simply opens a dimension by creating the summit line, the altitude, it creates area. is a dimension that envelopes degrees of intensity to degrees of extensity in one single intuition,

the intuition afforded by the area of a geometrical representation. Jake, are you speaking right now? I was, I was glad I had forgotten my question briefly and I remembered it as you said that. So is this another angle on your critique of like computationalism as adequate to a theory of knowledge? No, not anymore, I don't think so. No, no, no. No, no. No. Because I've never heard someone use the language of bits to describe ordered pairs,

like a bit of this, you know, of time and a bit of distance. This, yes, I know. That's cool. Okay, should we have a very, very brief moment of break and some water and stuff and then come back? That sounds good. How about five minutes? Yeah, sure, superb. Okay.

Thank you. Thank you.

Thank you. Thank you.

Thank you. Thank you.

I'm back. Thank you.

Should we continue or should we wait a couple minutes now? If you want to say something, ask questions, recall. Yeah, I just had in the sidebar the ability to imagine these sort of vectorial points, obviously seems like topological intuition seems very very important here it is geometric intuition or you know kind of a rudimentary topological intuition that idea of a co-exacts properties of the Euclidean diagrams at

least in in Oreen what you see that that was I was talking about the The conceptual basis of Aurim's insight is being preserved. What is not being preserved in modern mathematics or contemporary idea of, you know, integration of vectorial spaces, namely everything that basically happens after Riemannian geometry exacerbated by Katran's idea of mobile frame. You should check. This is basically Carton's mobile frame, Henri Carton's mobile frame. What is not being preserved is really the intuitive aspects of this, because mathematics

is becoming more and more autonomous in the sense that you no longer need the intuition for making these leaps, but simply formal relations that has already been derived from basic, what you might call intuitive axioms. Logical relations are not important rather than simply intuitive relations between the elements of your mathematical system. And this is really, you can see this,

that as I mentioned that mathematics can be thought in the most reductive sense and also in the broader sense, sense being driven forward by two vectors, the vector of construction and the vector of proof, the vector of mathematical discoveries and the vector of mathematical formalization, namely making mathematical systems. seems that the difference between modern mathematics and by modern mathematics I mean anything that's happened from 18th century on 21st century the difference

between this and you know late antiquity to late Renaissance mathematics is that mathematics no longer needs the intuitive component to advance. It can create its own intuitive components and that's really the interesting part. When we are talking about mathematical intuition in modern times we do not mean me, your or someone else's intuition in the Kantian sense, we mean a whole different kind of intuition and intuition that is based is no longer sensibilities in a Kantian sense or perceptual mechanisms but

mathematic relations between mathematical structures that can be generalized further and further and creates basically demarcate the development of mathematics in one direction or another so and this is really if you think about this this is all this is really the idea of intuition in Kantian sense so you have two idea of intuition one is which is quite rudimentary, the intuitive in Kantian sense. It's just basically what you might call to be purely sentient, subsapient idea of intuition, which really intuition is at its base. And the other

one, an intuition that has been driven by pure concepts of understanding. An intuition that has been equipped with concepts. This is not really the kind of intuition that we see the common understanding of intuition to be representing. This is what Kant calls productive imagination. imagination is simply understanding in camouflage in a Kantian sense understanding works with pure concepts of pure concept for categories pure

concepts of understanding are ultimately judgments judgments that can interrogate it's the limits of that parochial intuition, the intuitence, namely a form of intuition that is simply being derived from the byproducts of sensibility and perceptual mechanisms. Productive imagination as understanding in disguise is the one that should be seen as, yes, the intuitive component of modern mathematics,

but not that parochial intuition that is simply always changed the evolutionary basis of perceptual mechanisms. So you see, and this is really the idea of transcendental doctrine, mathematics is transcendental in the most hey you know emboldened hegelian sense in the sense that I was mentioning that you know it's a difference between a coffee machine and a hipster who is good at tasting coffee. A hippocere who is good at tasting coffee, namely who can

experience by way of a perceptual mechanism the taste of the coffee, is capable of making a well okay coffee. A coffee machine, who is driven by the transcendental power of concepts, now modeled on algorithms, is capable of of making great coffee without even experiencing the taste of a good coffee. You don't need to know how good coffee tastes in order to produce the best coffee. And this is really the whole idea of the transcendental difference between understanding, namely productive imagination, and intuitive imagination

that is purely based on perceptual mechanisms, the parochial aspect of experience. Yeah, I think this constructive vector, the productive vector that you're talking about, somehow, I'm not sure how to exactly articulate it right now, is coming back to this question I have of the axiomatization of the theory. Go on. Let me think about it for a little bit because I think maybe I'll put something more in the classroom or I'll think about it a little bit more right now, but it's like a little bit premature. Okay, sure, sure. Take your time to formulate the question. I mean the whole task of philosophy is formulating the best questions, not giving the best answers.

it seems like neurologically that this functions something in some way equivalently blind sight which is the phenomenon where your brain blind you know like your visual processing centers don't work but the hindbrain has a dedicated channel to the optic nerves and so like people who are just brain blind some of them at least you can throw a ball and they'll catch it or they'll get out of the way or like stuff like that and it's just like not they have no perception of seeing it their brainstem moves and they're just like I knew where to move and I did like is the usual after explanation. So it's like similar like we don't see we are like our perceptual mechanism is blind to the acceleration which is represented by like a block on this page.

Yes, but I mean, you see, the most trivial parochial example of it, which I think really the most genuinely beautiful example of it, is the difference between the computer and the stupid, junky, messy brains. you see the whole any person who starts the discussion from the perspective of experience alone is doomed to fall in the whole idea of pro-chialism of basically perceptual experience This is ultimately the doctrine of transcendental idealism. In fact, in order for us to have experience and to attribute it to ourselves, you need

to have concepts. And these concepts are indifferent to your experience. There are no ones. Yeah, no, absolutely, I'm just processing that. Yeah, that's exactly it. So it's a kind of possession where systems other than the mechanisms of conscious perception are using conscious perception in order to do the things that they need to do, which is like mathematics, which is figuring out non-intuitive elements. or not intuitable using a human perceptual apparatus elements in some situations.

Yeah, absolutely. And this is why I think that people really underestimate the power of Kant, and particularly Hegel, in terms of before all of this neuroscience came, before all of the... And I do believe at this point neuroscience is really the spearhead of the ultimate humanistic parochialism despite all of its you know claims to be on the other sides I think artificial intelligence yes artificial intelligence claims to be on the side of the transcendental ideals transcendental doctrine of knowledge and experience

But I think that with regard to people like Kant and Hegel, particularly Hegel, because Hegel is the one who sees really that understanding of reason are ultimately different beasts, far removed from the doctrine of experience as exposed by Kant. But nevertheless, Kant and Hegel gives a much more, what you might call, to be corrosive account of artificial intelligence, its ultimate vocation, that today's most, what you might call to be nihilistic, ideas of artificial intelligence, you know, proposed by the likes

of Bostrom, Lange, and so on and so forth. But if you really understand that basically what Kant and Hegel or even Hosserl are talking about, particularly Hosserl's Doctrine of Epoche, bracketing, phenomenological bracketing, transcendental phenomenology, you see that this is just like, it's quite scary in the sense that, as I mentioned, it's a difference between a superb mechanical coffee machine simply using the transcendental algorithms versus a hippister who is so adept in experiencing a stop in the world.

But just to the Bostrom and land crack, right? Because I just got to, like, if the underlying idea there, and this is purer for Nick than it is for Bostrom because of his whole... No, no, no, when I bunched them together, I didn't bunch them together in terms of that they have the same idea about AI, but I was simply saying that the underlying commitments, and what is this underlying commitment? The enmity toward idea of transcendental rationalism or transcendental idealism. The doctrine of a prior. That's fair, I mean I think, yeah, particularly each of them goes like too far towards that,

in particular, underline just the basic idea that recursive self-improvement, like in a context with accelerating access to computational resources, is like the whole distinction of reason from understanding in the movement from Kant to Hegel, not one of like recursive self-improvement as something that like accelerates and takes hold of itself, hold of itself and like acquires autonomy over time. Like I totally agree with like the, that artificial intelligence being something that like already exists in advance and it is the transcendental structure that we're using and that math develops through and all of that, like that gives a much less impoverished concept of AI. But that this sense, the idea of super intelligence as recursive, putting basic abilities

for recursive self-improvement into a fast time context leads to explosion. Yes, I mean, today's AI is simply an economically, technologically, but also cognitively affordable, namely a poor man version of transcendental idealism. Okay, I buy it. But the project of transcendental idealism then can only be completed by AI. I think the point is this idea of how we can basically create a project in which we are

capable of contrasting the achievements of AI as being instantiated today, you know, via computer science and neuroscience, with the actual transcendental philosophy of sapience as being instantiated within the so-called homo sapiens sapiens. This is a project that you need to understand this in terms of how this one can disbalance the other one and how the other one can disbalance the other one so as they create a higher point, rather than simply AI to be the spearhead of transcendentalism.

Right, because that's the account kind of of like how the curve actually happens. Like how does AI get better, like really make step changes and how good it is. Well, we invent a new way to do it or we add something onto it. And then there's a parallel line in which it then is able to solve problems much faster, improve itself much faster, but it goes back and forth. And so this dialectic of them imbalancing each other is how actual evolution in the intelligence available to each occurs. Yes, and from this broader, most broader perspective, you can see that AI is ultimately nothing but a project of fundamental alienation of human in itself. It is no longer an alternative, because there is no alternative to a human, sorry.

anything that you are going to come up with is going to be modeled from what you already have, namely your transcendental resources. But how are you capable of destabilizing your existing transcendental resources? Yes, that's a good question. Yeah, no, I, well, I have no immediately useful response to that. That's awesome. Thank you. Okay, let's get back to Oren. So as I was saying that... So I read that quote from Oren.

A couple of pages later in configurations, Orym continues by saying that consideration of these lines naturally helps and leads to the knowledge of any intensities. The one that we mentioned, that is basically he gives a cogent basis for the idea of modern intelligibility, how to render intelligible an intensive quality. The idea of measurement, basically, by establishing a dimension that can grasp the correspondence, the correlation between intensities and extensities, qualities and quantities in one single grasp.

Orin stresses that the line of intensity is not actually extended outside of the point, but is only so extended in the imagination. By this imagination, I mean productive imagination, not imagination in its lower Kantian dimension. And he actually uses the word, the phrase, secundum imaginum, second imagination, an imagination that no longer uses perceptual mechanisms but is equipped with concepts.

So Orym says that line of intensity is not actually extended outside of the point but is only so extended in the imagination and it could be extended in any direction whatever except that it is more fitting to imagine it as standing up perpendicularly on the subject informed with the quality these lines of intensity he continues ought to be called the longitude of the quality but generally and conventionally they are called the latitudes of quality now the extent of the quality in the subject if the quality is imagined as existing in a line

is to be imagined as a line on which the lines representing the intensity from point to point of the subject are to be erected perpendicularly. This line representing the extension of the quality in the subject, which ought to be called its latitude, is conventionally called its longitude. So the whole quantity of any linear quality is represented by a surface, a surface consisting of all the intensity perpendiculars erected on the

subject line. So again, and this is the chapter second of the configuration, Oren again makes this remark that the intensity lines should stand perpendicularly on the base or subject line of the figure for otherwise the intensity and quality would be laterally outside subject so you see I'm trying to

tell you why is that he comes up with this geometric representation why is that he should presuppose that the geometric representation of qualities within a subject needs to be understood as perpendicular lines erected on a horizontal line precisely because as he mentions otherwise the intensity and quality would be outside the subject if we grab geometrically represented as laterals as lateral lines rather than vertical lines

After all, we are trying to render intelligible qualities of a body with understanding that the extensible body is the only thing that is ultimately intelligible, the rift between modern science and magic. That is, so for otherwise the intensity and quality would be outside the subject, that is, they would be outside the subject both with respect to the altitude, the summit line,

which is, which Orain prepared to posit in order to have a two-dimensional figure represents linear quality, and with respect to the width or extent of the subject, namely the individuation of its qualities or its variations in time. Orym rejects, apparently, the latter, because he feels that the more immediate correspondence between the width of the figure and the extent of the subject would be better preserved by having the intensity lines perpendicular to the baseline. His conclusion then is that no quality is to be imagined by a surface or a figure having

an angle at the base greater than the right angle. It is as if he were trying to preserve some element of physical representation at the expense of achieving greater geometric generality. And indeed, he apparently achieved such generality by abandoning, or at least not specifying the necessity of perpendicularity. In sections 6 and 7 of Configuration, Oren then delineates his crucial and important

suitability doctrine that I mentioned briefly. Not any quality can be imagined by any figure. Rather, no linear quality can be imagined or designated by any figure except the ones in which the ratio of the intensities at any points of that quality is as the ratio of the lines erected perpendicularly in those same points and terminating in the summits of the imagined figure." That was a quote from Ory himself. When the figure that fulfills this condition is constructed is said to be

proportional in altitude to the quality in intensity and therefore the quality is most suitably or most fittingly designated by such a figure thus So, in the figure

A, B, C, D. In the figure ABCD, the ratio of FE

to Hg is as that of the intensities at the point E and G in the subject. and the same thing is true for other points and their corresponding lines, then the figure ABCD is a fitting one to designate a quality. Now, the chief consequence of this doctrine, the so-called doctrine of suitability, Noreen,

is that not only figure ABCD, but any figure on the same base, Any figure on the same base. And what was the base? The horizontal line. The extensible body, the subject. The chief consequence of this doctrine is that not only figure A, B, C, D, but any figure on the same base whose ordinates preserve the same ratio is a suitable representation of the quality of that body. Thus, any lower figures, such as, for example,

ABMN. AB MN. ABMN. Therefore any lower figure such as ABMN whose ordinates OE and PG on points E and G and all of whose remaining ordinates preserve the same ratios as the intensities would also be a suitable figure that is if GH the

ratio of GH to EF equals to GP to EO the ratio of GP to EO and similarly in the case of any other pair of corresponding altitudes of figures ABCD and ABMN, then ABMN is also a suitable figure to represent the same quality that the figure ABCD represented. is the same would be true of any higher figure as well such as for example a

figure such as a a b k l whose altitudes on any two points preserve the same ratio as those on the corresponding points in a b c d so you see our aim here is now making this very innovative way of uh combining the doctrine of ratios

and the doctrine of geometrical analysis by virtue of combining these two is capable of showing that you can have in fact multiple representations namely multiple intelligibilities of the same quality as long as they correspond to your baseline a b as long as they cover the same extensible properties you can see this is basically a fundamentally revolution thought in mathematical physics

you can represent the same for example particle field by way of the same geometric method different geometrical models it's an eigen space right each of these lines these lines are eigenvectors it's invariant under or what we're presenting with the summits is invariant under the scale or transformation yes yeah absolutely and these invariances are what the way that are being articulated here is by way of ratios applied to analytic geometry so does the representational schema of the diagram as the whole is an eigen space or is

like it was a yeah each diagram or each usage of the representational space that he's created is an eigen space of like physical interaction go on so far as making it so sophisticated for brain physical interaction but yes in terms of right one single body or one single body for one single subject yes that is absolutely true the suitability doctrine is reiterated as or in describes the

figures representing the basic kinds of qualities now let's alter the order on or in exposition and start with a uniform quality which he treats in section 10 a quadrangular quality it will be immediately noticed that the room adopts a shorthand method of naming the qualities by the figures used to represent them the first kind of quadrangular qualities a uniform quality

represented by a rectangle this a parallelogram the quality that is uniform in intensity is obviously represented by a rectangle since the altitude the line of summit this one since the altitude of the rectangle is uniform throughout uniform throughout

geometrically being represented in the vein of the parallelism postulate which we just talked about this and this namely the perpendicular height above any point of the base is the same applying his suitability doctrine, Orien further concludes that if equality is represented by a given rectangle ABCD, it is also represented by any rectangle on base AD.

on the base AB, assuming that AB represent the extent of the subject. This is obviously true since in any rectangle or the ratio of the altitudes, since in any, sorry, since in any rectangle, the ratio of the altitudes on any two points is a ratio of equality. oops a B equals to one the only provision in the arbitrary selection of the altitude of the rectangle is that we are to compare two different uniform

qualities in equal subjects say one that is double the other in intensity then And the doubly intense quality must be represented by a rectangle that has twice the altitude of the arbitrary selected figure which represents the initial uniform quality. For example, let's geometrically represent, grasp this in terms of ABMN and ABCD. The same considerations enter into Orem's discussion of uniformly deformed qualities,

namely qualities in which the intensity of the subject quality varies in a uniform way as it is distributed through the subject. and what was the graphical representation of uniformly reform variation what is LK oh no no in terms of the the most broad it was a right triangle So, Orym discusses two categories of uniformly deformed qualities.

The first one is a uniformly deformed quality that begins from zero intensity and ends with a given intensity, or which begins at a given intensity and ends with zero intensity. is exactly a right angle which by the way or in is responsible for as I mentioned he came up with the basic idea of classical perspective you see a right angle the geometric representation of that he has started to compare it by way of the diagram between mechanics machining engineering and geometrical analysis that I mentioned in previous session he was capable of going back and

forth between the studying of ax uniform acceleration on an inclined plane and the geometric analysis of the area of right triangle for uniform acceleration then he was capable of taking the convergence between the mechanical inclined plane and the right triangle in Euclidean systems and studied them under Alhazen's and Nathan's optics to show that this is exactly the case of classical perspective classical perspective if you think about it is

simply what you might call to be a three-dimensional expand version of an inclined plane or a right triangle namely a uniformly deformed motion in which quality begins from zero intensity and ends with a given intensity or begins at given intensity and ends with zero intensity what is really you see this is the whole idea that you can think of the observer of the classical perspective as a static observer zero degree

of intensity as it puts its body in motion it sees that the horizon expands the right triangle the 3d version of the inclined plane the expansion of this horizon the area of the right triangle that is now being diagram is the very idea of the classical perspective classical perspective is nothing but an optical version of how mechanics and geometry converge upon one another and if you want to get into details of how this can be possible in details check the

second chapter of chat less this one will be the stake of the mobile which is about ORIM, where he talks about how basically you can derive the diagram of the classical perspective simply from ORIM's diagrams. Because in classical perspective, I guess as you move towards the center of the picture, pixel by pixel or brushstroke by brushstroke or whatever the degree of like the degree of acceleration of the the degree of change in the ratio between the distance between brushstrokes and the distance between

objects represented in the painting grows so you're experiencing growth and acceleration as you move towards the center in order to mimic like seeing a horizon line is that yes yes yeah absolutely and of course they they didn't talk about in terms of acceleration what they were talking about in terms of variations of qualities within qualities for a single subject so basically what or indeed he established a correspondence between the geometric analysis of uniformly deformed motion namely variation of qualities between the laws of inclined plane that has been given by Archimedes

about uniform acceleration and the discoveries of Alhazen in terms of prospective operators which talk about the qualities of Lux light How would I mean maybe this is just like would go off like totally tangentially but I how would you represent reasoning about that in terms of reasoning about lights or like the movement of light properties of light because like just like during where we've

come at it so far that's not it no when we are talking about that's why I use the word locks because I didn't want to use the word light so you can get the idea today's mother theories of light when we are talking about locks we are talking about a different kind as a tea which is a conceptual a completely different it's like when we were talking about motion when we are talking about motion and scholastic philosophy we talk simply about a quality, an intensive quality, that can be expressed extensively. The same thing about lux. And Chartier actually again talks about, the whole chapter is about Alhazen's theories of lux and lumen. Okay, and then so lux is something along the lines of like the degree to which you can see, like visibility,

visibility, like the whole projection of light from God or something? Yes, from the eye, from the observer, yes, absolutely. Which again, being represented by analytic geometry. Right. Goethe, his theory of color perception or light is like projected from the eyes too, isn't it? Yes, yes. Alright, thanks. So as I mentioned, Orym discussed two categories of uniformly deformed qualities. The first one is a uniformly deformed quality that begins from zero intensity and ends with a given intensity, or which begins at a given intensity and ends with zero intensity.

Such a quality is to be represented by a right triangle. For this reason it is called on the title of the chapter 8, a right triangular quality. Orim is once more naming the quality from the figure by which it can be represented. Again, Orim applies the suitability doctrine and concludes that this quality could be equivalently represented without any difference by every triangle having a right angle on the base again you remember that we were talking about the doctrine that between the parallelism and non-parallelism there are infinite degrees of variations

that you can represent so we as you can see so we have this and we have pure perpendicularity pure intensity and between these there are infinite variation how you can represent this again a uniformly deformed quality by different source of triangles bright triangles so This is proved by showing that the ratio of any two ordinates on the base of a right triangle on a given base is the same regardless of the height of the triangle.

Therefore, some quality is assimilated to any of these triangles, and further, the same quality can be assimilated to any other one of them and be imagined by it. There is this provision, however, if some quality is designated by one triangle, another quality of similar but double intensity must be designated by a triangle that is twice as high and similarly for the proportionally great intensities orim then goes on to say that there are triangle triangular qualities that are not represented by a right triangle

but rather by triangle whose angles on the base are less than right triangles these are qualities composed of two uniformly deformed parts one beginning and one ending with zero intensity and the suitability doctrine may be equally well applied to these compound qualities and this is where he gets into his doctrine of deformally deformed qualities what you might call to be non-uniform acceleration in modern times we are represented by curves of this form whereas orim represents them as diagrams some of this kind so you remember uniformly uniform

uniformly deform and this is deformly deform sorry and you can think about this idea that the reason that he represents with triangle has something to do with the doctrine with the thesis about exhaustion particularly the Archimedean version that you can exhaust any curve simply by drawing triangles you can

approximate any curve of this kind by triangles of this kind. So now as your homework and kind of thinking exercise, according to this suitability doctrine the way that we can represent qualities by different diagrams by different geometric representations think about how uniformly uniform

variation of qualities or motion can be expressed in terms of a uniformly deformed representation was the present representation as we said is a right triangle the hint is that so if this is uniformly uniform speed or velocity it can be represented by a uniformly deformed velocity, namely uniform acceleration.

But you need to articulate under what condition can we say that the rectangle and the right triangle are equal hence they are both representing the quality of the same subject but within different variations using to express this you need to make an assumption about these two geometric figures their areas how they should basically intersect with one another can you explain a little bit more why it's the same subject first of all they are really covering the same subject as you

see there are both on the same base X right but that is not that is a necessary condition is not sufficient condition for uniform accelerat, uniformly deform diagram to be equal to the uniformly uniform diagram. For uniform velocity to be corresponding with uniform acceleration, with mean speed be corresponding to uniform acceleration. You need to make another assumption in this geometrical representation in the vein of Euclid's elements. So then be capable of in precise analytic geometric terms like Euclid's proofs showing

that under the second condition, given the fact that they are already sharing the same base, the areas can be set to be equal. they are both these both diagrams are equal and that if you can come up with this then you I will argue that this is exactly the formula or the fundamental insight that Reims developed to elaborate a relation between meanness speed and uniform acceleration that ultimately put the Galilean revolution in motion so

ultimately we are talking about a rectangle and a right triangle sharing the same base this is their necessary condition they are both covering the same subject but we need to add another condition to this geometric representations for the rectangle and the right angle the areas of them can be said to be equal I'm sorry, did you ask a question just then?

No, no, no, I was just like elaborating the problem, the homework problem. have a necessary condition come up with a sufficient condition for these two geometric figures can be said to be equal and then in the vein of Euclide proof show why is that they are equal even that sufficient condition as I said once you do that then you see that if this is really the case under this second condition, the sufficient condition, we can see a relation, an intrinsic relation between uniform acceleration and mean discrete.

How much more time do we have? I'm sorry I love it looks like we have 20 more minutes still okay good good so that was a bit on a ring as I said you can read there are so many great books I will put some of them in the in the Google Drive or something really great books written on Aurim and his contribution. But nevertheless we are going to stop our commentary on Aurim and the doctrine of quantities of qualities aside and move to another great

example of how analytic geometry advanced the development of mathematics, particularly algebra. I'm going to give simply the method of Al-Khwarazmi here and then I will leave all of the introduction to this basic themes of algebra for the next session. So I assume that the majority of you know what a quadratic equation is. Who doesn't know what a quadratic equation is? So we can shave them. I'm just kidding.

So very briefly, quadratic equations comes in this form. Now, this is the reason that quadratic equations are famous because is that if you look at this equation you see that there are some stuff here that we haven't had so far we have zero one part of equation so we have introduced something a new

entity called zero we have also squares and other than squares we have also just simple unknowns and we have also constants this is C constant something that doesn't change throughout our equation now quadratic equations are the most elementary components of of elementary algebra. Historians think that the first person who fully elaborated on the nature of quadratic equations,

hence incorporating the importance of zero constants and squares into mathematics, was Al-Khwarazmi, the father of algebra. But, recent historical studies show that in fact his solution wasn't so original. It was already discovered 600 BC by Indian mathematicians. So how are we going to solve a quadratic equation?

If you look at the quadratic equation, you see that we have something here, x power 2. What is x power 2? If you see x power 2, it's x times x. What is this? This is the area of the square whose sides are unknown. So the solution to the quadratic equation is also that was given by Khawr Azmi and as I mentioned was in fact given much earlier by Indian mathematicians is also known as

completing the square because quadratic equations are methods of squaring. Quadratic simply means to square. There is this manuscript discovered apparently by, I mean, I think in the 20th century, around 1960s, 1970s. It's called, the Latin name is Sul Bastron by Budiana, which is, dates back to 600 BC.

It gives a very, very, very interesting method of geometrically representing squares and anything that can be expressed in terms of squares, namely x power 2. The solution goes like this. So you see, again, back to before moving too fast.

that this quadratic equation, from a geometric perspective, is the problem of an unknown rectangle that has been decomposed into squares and some remainders. Squares and some remainders. Ultimately, to solve a quadratic equation is a problem of how to turn a rectangle to a squares.

The Indian manuscript gives this solution. an oblong into a square by taking the shorter side of the oblong for the side of the square. So you have some rectangle. one take the shorter side of it X cut the same amount from this oblong as X as a

shorter side of it separate this you have a remainder now divide the remainder Namely, the part of the oblong which remains after the square has been cut off into two parts. Divide this oblong to remain into two parts.

Now, inverting the places of this oblong that you just cut to two equal parts, join these two parts to two sides of this square. So this was phase two. three is that you have you had your square and we know that these are also X so you get this part bring it here this X sits on this you bring this one

and this x also sits on this. Now connect these lines together. We know that whatever this was as z for example we divide by 2 so you get z z2 and z divided by 2 so you have turned a rectangle into a square Now Khara's me simply uses this geometric reconstruction and comes up with the method

of completing the square. It is the solution for your quadratic equation. Now, how does this work? Let's have something like this. Oops, sorry. x2 plus 8x plus 16 equals to 0.

So the first thing that we are going to do, what are we going to do? We are first creating a rectangle and our square. This is the square and this is a rectangle. One side of this is x. You see the area of a rectangle is one side multiplied by the other side.

X times 8, 8x. The area of a square is 8x and x2. So basically all you need to understand, as I mentioned, that the quadratic equation is you have a rectangle and you have a square a rectangle and a square given

the geometric solution that I just gave how can are you capable of if you have this and this and this constant is okay you can we don't want at this point talk about it because we are talking about unknowns the X's so we have X power 2 and 8x namely the process of how we can make a new a square from this X to an 8x so we have to divide 8 to 2 in accordance to our method of dividing the remainder of the oblong to two we

have our square then we put four so this divided by two is four and put another parts of this here we know that these are X this is X this is 4 this is 4 this is x and this is x now we connect these together what is the area of the small

the square the the additional square that we just made 16 or times 4 from this you can go on and basically solve any quadratic I want to leave some subtleties of how you can solve really an equation with this but let's go with something like this again as your homework so solve something for me like

remember one help that I can give you when you are constructing this new a square which was now the area of it was 16 do not forget to make the additional change to also your constant you have introduced something here which also should be manifested in your constant that change you can't simply leave your constant unchanged once you construct this new square so x2 plus 10x plus 39

equals to zero. Using the geometric method of completing a square solve this quadratic formula. again just as a kind of a recap method the Indian method is like this you have an oblong let's say this side is called X the shortest side of the oblong you

take the shortest side of the out long cut the same amount from the longer side create a square producing a square through this procedure then you have You have also a remainder, it's called this y. Second stage, divide the longer side of your remainder by two. Cut it half. This is also x.

Then, in the third stage, this is your square next to bring this here this is why to this side and bring this side here I'm going to create a square this is why to the area of it

so you see what is really an X square or powers basically you are what you are doing is you are reconstructing powers in the fashion of war II you know or in basically allowed you the graphical representation to by way of combining ratios and analytic geometry to express different graphical representations of the same quality within the same and one single subject what you are doing here

here is similar to that you are decomposing your qualities or your properties and you are recomposing them back in accordance with the most basic single element that you can intelligibly articulate it and what is that it's your square you have an oblong you decompose it in a way that you created a square out of it you took this square put it one way then you have your remainder then this remainder divided by two so it can be glued to this square and once

Once you glued it to the z square, now you can render it intelligible according to your basic intelligible element, your quadratic x2. Where most intelligible here approximates to like takes the least amount of information to represent or most manipulable, like least steps to manipulate? I think most manipulable, the one that you can simply take it apart and reconstruct the

entire remainder of your system according to this component, which is the square really. decomposing, recomposing according to your most basically manipulable, rudimentary, simple, intelligible component, which is the square. Cool. Was it of like the way in which the rectangle was a generalization

Quadrilateral of a rectangle and so forth like we're moving in from like the more generalized rectangle to Like it's most packed variant or it's most compact Yes most yeah, most most invariant in terms of that You remember, actually, for the Greeks and also for Indians, a rectangle does not represent the generalization of a square. It's from a modern perspective. It's a square that is the general of the rectangle. A square is the perfect shape. So you are, it's like really a pure form.

So you are articulating something that is impure by way of a pure form, namely a purely intelligible entity. Okay. Thanks. So that was today's session and next session we will talk a little bit more of this and we will talk about our new pair of concepts, balance and disbalance. I have a question or at least a riff I wanted to throw out going back to the dazzling conversation

between you and Jake, which roamed from Hegel and Kant to mathematics and so forth to AI, which was really stunningly interesting. What I was thinking as I listened was there's a tendency, one thinks of analytic geometry at whatever point one wants to introduce it, but let's just use the name Descartes for the moment even though we have intimations of it before, as creating a toggle switch between numbers and figures. One can simply go back and forth between the two, between arithmetic and geometry,

and until one leaves three space and goes into higher spaces, which are entirely numbers, and one is out of that. But there's a great deal of pleasure in the toggling back and forth between the two. If you're looking at transformation groups and you go from a page which is showing the rotations and reflections of a triangle to let's say a number thing which is a permutation group. And it's kind of fascinating to see the two sides of the toggle align in your mind as you're doing it. But the vision I had as you were talking about AI,

is the notion that this synthetic ladder, the synthetic part of mathematics is a ladder which takes us up to a point, or takes the mind up to a point, where something, let's call it AI, AI is waiting in the universe, which is all of that mathematical activity or mathematical latency that doesn't require figures at all, and in which the largest portion of modern mathematics lies. so when you open a book on like projective geometry and you might be looking for at least something that starts with perspective and painting and and so forth you know you just

have a projective plane defined with a number of axioms and everything goes from there and there's not a figure at all in the book which is fascinating so uh uh on the one hand i'm wondering what happens to diagrammatic reason as it was outlined in the beginning, the notion of pop-ups and the notion of this other form of reason that's working with the drawings and the diagrams as you go along, but which seems to have entirely disappeared at a certain level. On the other hand, in a totally unrelated riff, I just happened to be reading these wonderful science fictional novels by Matthew Diabetua.

I don't know if anyone's ever read them, The Red Men and the Destructives, which is about a version of AI. So I was riffing in the end on the notion of maybe something close to what Penrose and Tegmark say of mathematics as something that's out there in the universe that we can call either AI or mathematics. And that human beings, as they say, have to follow a ladder that's synthetic to get there, but can't even go into the farther reaches of it because it's just out there in the universe and it's the next post-human stage, as you might say. Regarding to the diagrammatic reasoning one, I think diagrammatic reasoning one can be,

I mean, we can also understand it in terms of how we are ascending this, what you might called to be the support ladder, the Jacob's ladder. In the sense that, yes, in the Euclidean sense, it comes in a very rudimentary type, but as we are moving up the ladder, also the pop-up function, but also our diagrammatic reasoning, also become more convoluted in the sense of exactly between what kinds of entities we toggle. but even at the most basic level the diagrammatic reasoning and functions is in fact toggling between concepts deductive concepts and deductive logical relations and intuitive perceptual

components we talked about it yes we have for example something like the definition of a point its limits are the limits of perceptual mechanisms. But we said that we can't really talk about what a point is or what a line is, what a square is, what a circle is, unless we pose it in a logical relation that can be deductively made explicit with regard to other geometric definitions. But I guess what I was saying is that AIs don't toggle. And after- AIs don't toggle, but nevertheless, their toggling, I think, needs to be thought in

terms of a higher, as I said, different types of pairs of entities. And this is really, as you see, it's quite a familiar Kantian trope that this, what you might call to be the synthetic Buddha strapping is really of how productive imagination and pure understanding play on one another to the point that imagination is being fundamentally morphed by understanding but is it still capable of pointing understanding to new orientations understanding to new orientations, capable of disbalancing understanding. No, I see that in Kant.

And what I'm asking is, or let me frame it as a question. Do you think there's a point in mathematics where people who are doing it stop toggling? I know I'm at a point where I toggle. I mean, I find it enjoyable. But I'm wondering if at a certain point, you know, people doing mathematics don't toggle anymore. That would be the question. I would say, I mean, any answer I think to this would be purely a speculative, particularly from someone who is not by any means a mathematician. But nevertheless, I would say there will be a still toggling, but as I said, toggling between different types of mathematical entities, as today is really a witness to that whole

whole idea that in order for us to expand the landscape of mathematics, the toggling is no longer between simple mathematical components and simple intuitive components, but between empty elements of mathematics and elements of logics and elements of computation. Okay, I do see that then, because a lot of mathematical discovery, or so-called, is about people working in one area of mathematics and somehow making a connection to another area of mathematics that no one has ever seen before as being related or part of it. So I guess that's a form of toggling that continues in mathematics.

Yes, but not in the sense that, as I said, it's not in the sense what you might call to be the pure polarity of how perceptual intuitive components are being contrasted with you know components of the understanding of pure concepts now we have different levels of semantic concepts are being pitted against one another and the second one I think the Panbrosian scenario is, I mean, obviously from a trivial, superficial perspective,

it is platonic, but I genuinely think at a deeper level it's also purely platonic. This is ultimately a platonic position that intelligence is purely a teleological entity, but not teleology in Aristotelian sense of causality, final causes, but a teleology that is fundamentally infinite and transcendental in the sense that intelligence is capable of coming with its own telus and expanding it. Great. I have a question.

I would like to hear your response in this matter. We're talking about land and intelligence. It's very interesting. It approaches the problem of AI, but it's more... If we think of the intuitions that constitute the understanding, I do possibly agree with Land in saying that they are materially constituted. The constitution of the understanding is effective because it's materially proved its efficacy. Except my, what I'm seeing at issue with that though, it's what you're seeing with the whole transcendental problem.

And like, I want to know if you think like the trans, like, it seems to me like the transcendental is material, like, it has like, the reason why the transcendental works is because there's a certain pragmatic efficacy of the transcendental itself. I'm not sure if that's the wrong way to look at it or what you think. I think this is really the difference between Hegel and Kant. And that's why when I was talking to Jake, I said that I particularly emphasized on Hegel rather than Kant. I think this pragmatic efficacy is unfortunately the conservative aspect of Kant, where basically

reason is reduced to understanding, and in so far as understanding is essentially rooted in what you might call the pragmatic efficacy of being thus and so constituted by what you might call to be the causal substrate of material this reduction becomes inevitable but this is only in so far as the reason has been already reduced to understanding and we know that pure concepts of understanding first of all I mean I think Nick doesn't have a very good grasp of this aspect of Kant the pure concepts of understanding, which are transcendental categories, are not really

derived from simple, pragmatically charged instances of experience. Namely, they are not in fact derived from experience. This has led to this really bad misreading that transcendentals are some sort of supranatural components or entities. No, they are derived from experience, but not from particular instantiation of experience that you can track them and explain them within particular causal agenda. How they are derived is that they are derived from the method by which mind puts together

and generalizes particular instances of experience. They are derived from the manner of organizing particular forms of experience rather than just from disparate different instances of experience. So that's one thing that needs to be taken aside. So even when I'm saying that reason is irreducible to understanding, I do not mean that understanding can be reduced to particular experiences and particular causal mechanisms. But nevertheless, yes, understanding is narrow to the extent that is rooted in what you might

called to be is pragmatic efficacies and this is really any person who is familiar with the categories knows that categories of for example time are essentially bound to for example how memory is being instantiated or how for example the categories of causality again bound to the structure of the memory in the doctrine of inner sense but reason is different thing is a ultimately a different beast altogether than understanding and the categories of understanding yes are parochial and in fact I think they are susceptible to

complete revision so but the great thing here is that precisely because if we are changing the categories we admit that categories approach your pro-choal ultimately in so far as they are basically you know balance these causal material patches if we revise categories we are also pointing to the limits or of us talking about the nature of understanding and the nature of reason based on the current material substrate that has instantiated these mechanisms

of understanding and rational faculties. So it's a double-edged thing that if you really say that understanding is tethered to these material substrates and because of its parochial, then if we are capable of, by the power of some other faculty, capable of renegotiating the limits of understanding, pointing to the new pure concepts of understanding, then we are capable of thinking about different material organization that can instantiate new forms of understanding. Which means that by virtue of that we are also pointing to the limitedness of thinking,

essentially a parochialness, a narrow-sightedness of thinking about understanding simply in material terms. material terms that we are talking about when you're talking currently about understanding are they're already realized material substrates this doesn't mean that understanding cannot be realized by other material configuration material organization that can lead to different kinds of categories. So what you're saying is that while the revelation or construction of Platonic forms has a sort of naturalized particular revelation, its necessity does

not derive from its natural particularity? Yes, yes, and this is fundamentally Plato's and Hegel's thesis yes absolutely and a great really a great any person who is really interested in this diffusion of this Plato and Hegel with regard to the nature of intelligence particularly the doctrine of transcendental and how it is really different from these particular instantiations material particular material instantiations it's John Neymar Finley particularly is trilogy let me write it

down here I thank you Would it potentially be accurate to say that reason here then is the condition of possibility for complementarity between the evolution of the understanding of the material substrate? Yes, and then you see that why is that even in the most absolute idealistic theses of

Hegel, you essentially see that the transcendental, basically transcendental idealism, and the same thing with Plato's Doctrine of Forms, is a thesis that can also be thought in terms of how to re-engineer material reality. And then Christian and I were talking briefly earlier, a complementarity between the transcendental reduction and the process of constructing some new way of thinking in the process. Is that what we call the production of imminent residues, the sort of like the combination of transcendence from a structure. Is that kind of...

Can you elaborate on this point a little bit? Jake, it's also really hard to hear you. You're cracking. Craction problems. Okay, you're good now, though. I'm just going to... Can you hear me? I can hear you now, but... We can't hear you now. Yeah, if you want to post it in the classroom, that'd be great. Let's try and start some more active threads in the classroom, perhaps. I also sort of wanted to ask about the role that Freedom plays in Hegel. I think that fits in here.

yeah I mean you can definitely see this is you're talking about this idea of the trans the power of the transcendental dimension the first thing that comes are both it's not just freedom but also what you might call to be antinomies of freedom antinomies of freedom in the sense that basically we have at the same time some sort of what you might call to be bondage to the material constitution or a material instantiation I think it's

instantiation rather than constitution and also we have what you might call to be the the autonomy of the transcendental components which are not about free will we will is simply what you might call it a deluded idea that tries to understand freedom in terms of material constitution and hence the idea of voluntary this also some other stuff to come with autonomy has nothing to do with that so yes for Hegel is how to basically resolve the antinomy of freedom by allowing to create a form of what you might call dynamic

this equilibrium that allows to move, to ascend, between the bondage to the material constitution and the freedom from the constraint of the transcendental entities, namely the self, the power of transcendental components to be capable of making their own roots, their own constraints. So he wants to move between, in a platonic sense, there is no such thing as freedom in terms of universals or freedom in terms of particulars. But ultimately the idea of freedom needs to be understood in terms of the bipolarity and

the tension between the particularities of experience, which are completely different from one subject to another, and the transcendental ideals, or in Platonic sense, the idea of concrete universals. And the reason that I mentioned Findlay, because he thinks that this antinomy cannot be overcome within the realm of current material constitution, what you might call in a Platonic sense,

you can't overcome this antinomy with the same limitations of the same body that you right now we all share, that you need to re-engineer this. You need to be capable of completely get rid some of its constraints because fundamentally there is an antinomy there no matter how much you say that we can in fact implement that kind of concrete universal freedom with the currently instantiated limitations uh that are basically out of which our bodies are being constituted you cannot really solve this problem and that really leads him to come up with a really excellent although mystical but not in a religious way rational mysticism solution

well we are at almost four o'clock now is anybody have final questions before we wrap up yeah I I have some questions if that's okay. Sure. It's okay with Reza. Yo, Reza, is that cool? Sure, absolutely. Yeah, so I missed the entire class. Like, I was there for a little bit session one. Now we're in session nine, which is like over time. Yes. So, like, I'm thinking about, like, protocols for patching programs. so like 0.9 versus 1.0 so like if this is class 9 and we're having one more class and that'll be

class 10 we have two more classes okay no I don't I don't even know how many classes I haven't even been here but I woke up got the email decided to join the class but here's what I'm thinking like in terms of like MMOs or like games, usually like 0.9 would be the patch right before 1.0, which is like launch, right? So I will definitely be here for class 10 and I'll try to do my homework. Okay, and the second question is about like triangles. Like right triangles versus left triangles. but like you know Mark Fisher?

Yes. Like his Facebook page he has 3,333 friends and I'm just wondering if you have any like algebraic explanations okay that's all I have that's all I have Go on go on algebraic explanation of what? like why does he have 333 friends huh and like I think mystery I think that he died on Friday the 13th under a full moon you need to consult

yes I think with a not only with a statistician but probably with the computational mechanics Yeah, that's definitely something probably with the Facebook algorithm. That's why I'm talking to you. Wait, I thought that you were a figure. No, no, no, I'm not by any means adept in these kinds of fields. Okay, excuse me, I must have had you confused for an algorithmic operative engineer. engineer no well I'm engineer but the kind of engineering that I have been doing has actually I've never worked with algorithms because engineer can also go with simply geometric diagrams without using algorithms honestly Reza

Reza you're probably not even smart enough to understand algorithms so don't even try I won't all right thank you thank you this has been a really awesome class this has been one of my favorites we got we got a lot of interesting like stuff yeah definitely I mean you should as Jake was saying at the beginning of the session definitely if you I mean there are a few people who are you know absolutely you can basically you can ignore the entire history of scholasticism and just concentrate on deeds. There are a few people in the, you know, what you might call to be the Muslim sphere of medieval scholasticism, Ebne Haysan, Al

Hayzen, Abyssinah, Al Khawrazmi, but in the realm of the Western realm, it's definitely boring it's Richard Swine's head and Brad Ward on basically Paris School and Merton College Oxford when you said Brad Ward on Brad Ward I I I wanted actually to talk about him but then I thought it would be just too difficult particularly for people who are not familiar with math because he is the one who comes up with the with the modern definition of logarithmic functions simply using ratios

was he that's um orange teacher yes he was or his mentor yes yeah absolutely who who did you say again the last one i got richard swine brad brad wardyne i think uh i i i have his uh i have part of his manuscript uh i can upload it to the google drive but if you want to read about these check this fourth yeah fourth volume of a book uh which which is considered to be the most exhaustive historical study of the relations between natural magic and development of science in Middle Ages by Lynn Thorndike. It's called

The History of Experimental Science and Magic, the fourth volume. All right, thank you. And one more question. Finley, I think he's the guy who wrote the preface to Hegel's Phenomenology. Yeah, the foreword. What was the trilogy that you were referring to by him? it's actually a four books but people but it really came in the form of a trilogy it's called the discipline of the cave uh the transcendence of the cave and the ascent to the absolute all right thank you but there's you said there's a four yes it's called and the fourth one

is called Axiological Ethics. Alright, thank you. Yes, his actually commentaries, I mean, if you want to read Hegel, particularly where he, because you know that the phenomenology of a spirit is simply breaking away from Kant, namely questioning the parochialism of understanding. But he does not talk about really the idea of transcendental reason in phenomenology of the spirit because as its name says it's phenomenology it's in science of logic and the best commentaries on science of logic has been written by I mean John McTaggart and Fiendley and also if you really want to see one of

the best critiques ever written on cans showing the inconsistencies within the Kantian system is the Hermeneutic of Transcendental Hermeneutic of Transcendental Idealism, I think, again by John Finley. All right. Thank you. I really appreciate it. I haven't, like I knew that he wrote the preface, but I didn't know he was such like... He's considered to be one of the best philosophers of the 20th century. Yes. It's really, it's quite really, really fantastic.

All right. All right. Yeah. I'm glad to hear that. Got some studying to do. Okay. Thank you so much, everyone. All right. Thank you, Reza. Have a great day. Thanks, Reza. Great session. Absolutely. Thank you. Bye-bye. Thank you.