Hello and welcome to the eighth session of Maths and Ideas with Reza Negrestani. I'm going to pass the mic off to him now. Thanks. Okay, so let's start today's session. I'm going to briefly talk about a very, I think, important, at least from a historical point of view rather than kind of a philosophy of mathematics perspective relation between mechanics machines engineering and mathematics of
of late antiquity to late scholastic era, and why is that we need to pay attention to this relation between machine engineering and the hegemony of geometrical analysis throughout this time period. After that, after this brief excursion, I'm going to talk about the work of Nicole Oren, inductively the most important natural philosopher of Middle Ages. He was an archbishop, an astronomer, economist, mathematician, magician, philosopher.
going to talk about a little bit about his his invention of mathematical graph and how he which as I will explain the invention of mathematical graph has something to do with this relation between analogic geometry and mechanical engineering namely the relation between the mechanical analog and the geometrical model and given the fact that historians of science now completely agree that Galileo's
and Kepler's formula for motion of celestial bodies were not utterly revolutionary but simply modifications of Orem's formula mean speed and acceleration that makes Orem's contribution that I will talk about even more compelling from a sort of perspective. The reason aside from Aureem being you know the most important national philosopher and logician or mathematician of the Middle
Ages, the reason that I decided to talk about Aureem's contribution is because I think OREEN consolidates some of the most important observations about the relations between mathematics and physics in the Middle Ages that led to the so-called Copernican Revolution and ultimately conditioned what you might called to be the modern framework for natural sciences whatever we know as
science at this point so before starting our course is there any question about the previous sessions about particularly the last session we're talking about Euclidean system and the internal configuration of Euclidean diagrams yeah I posted a question in that classroom after reading the stegmuller text that you recommended or I just read the chapter the axiomatization of a theory
And I was just wondering if it's not too much of a side conversation, if we could talk a little bit more about the role that intuition is playing. And then the more basic question that I have is how do we separate the constructive role that postulates and definitions play from the relational role that axiomatics play. they're just sort of really broad questions if it's too if it's too distracting right now then maybe it can happen okay just just very quickly remind me I read a segment was text a long time remind me what he what exactly do the difference between the constructive approach and relational
approach or the relational rule that's he only mentions that really briefly let me just pull it up really quick is this from uh his book on theoreticity this is from uh yeah the structure and dynamics of theories is that yes yes yeah and then it's in chapter two on axiomatic theories I'll just read sort of slowly this one paragraph because we are for the moment concerned only withdrawing a comparison with the Hilbertian notion of an axiom system we could just as well have spoken of the concept of an axiomatic system in the Aristotelian sense. Historically though various differentiations would have to be made that for the most part remain
neglected in philosophical discussions of this topic Existential quantification, for instance, plays a different role for Euclid than it does for Aristotle. Furthermore, axioms in the strict sense, relational sentences, would have been distinguished from postulates, construction sentences, an undertaking which presented considerable difficulties even in ancient times for very penetrating and concise discussion of these. And then he lists a place to go look. underlying the evidence postulate according to which the truth of all axioms must be directly apprehensible is the idea that fundamental notions employed in the axioms be derived from
intuition so that the evidence takes on the form of an insight emanating from these notions okay I think I mentioned this a couple of times that's I I know that people like Lauren's on told and by the way you absolutely have to read Lawrence Pantel structure and being if you are interested in this whole idea of what really philosophy is and should be, what is the relation between philosophy and sciences at this point. I highly recommend Lorenz Pontel's structure and being. One of the things that
is important here to mention is that even though I think people like Esegmuller and Pontel talk about Euclid's definitions as an intuitive axiomatic system, I really do do not think that Euclid definitions are axioms, even in the intuitive sense. But let's believe that they are axioms in the essay Mullerian sense, that they are derived from intuition, the Kantian sense. I think, so there is a, I think even, So from an intuitive, Kantian intuitive perspective,
Euclidean's definitions can be dealt with as what you might call to be rudimentary axioms. The difference between Euclidean systems and Euclidean axioms and Hilbertian axioms is that it's something quite specific actually in the sense that, so this is really a Seymour insight that when you have intuitive axioms in the Euclidean sense whose
limits of perceptual systems rather than limits of language or logic is that the intuitive the kind of system that you can build on these axioms is going to be a one model system whereas Hilbertian definition of axiomatic systems allow you to interpret any models namely cohesively finite and this final means any large amounts of models within your given axiomatic systems so why is that
it's precisely because how Bertian system is about this what you might call to be macro theoretical relation between the statements of your theory you can think of this as the same statements regarding your postulates they have been couched in terms of or in the vocabulary of natural language but also in terms of their relationships the relationships between and these relationships are not just any form of relationships or logical relationships between your axioms
that can be couched in set theoretic terms now Hilbert's Hylbert's picture of an axiomatic systems allow you to derive multiple models from one axiomatic system not because it completely abstracts away from the intuitive import of your so-called axioms but precisely because it sees an axiomatic systems across different scales in the sense of some
formal in sense of intuitive import of axioms name your self-evident building blocks of a theory then the relationships between these intuitive axioms which are usually logical and then at a higher level in terms of what you can construct the model that you can construct out of these logical relationships so from this perspective I think Hilbertian understanding of an axiomatic systems particularly with regard to the Euclidean system is not by
any means a form of rigidification as if you think that oh precisely thoughts of of Hilbert's school of formalism is going is basically proposing some sort of complete break away from the intuitive imports of euclidean axiomatic system no that's that's really actually that can the couldn't be far from truth in fact even even though Hilbert goes for the formalist approach namely trying to talk about an axiomatic system in terms of relation life of relationships between your axioms allow you to build
the different models he basically from a certain perspective is arguing that even though you clean systems is a completely intuitive systems in terms of its axioms or definitions if you manage to be a stratified or or sorry if you manage to hierarchize basically the study the system along different scales the axioms the postulates the constructions the proof so on and so forth logically in
in terms of logical connections or relations, then you are capable of in fact developing different models out of a Euclidean system. And some of these models might not be even compatible. So this is I think a really, an interesting insight that simply by allowing a hierarchical view from a logical standpoint of Euclidean system, and what axioms are and how they stand in relation to one another, and how by virtue of these relations they afford certain constructions. Hilbert argues that if we take the idea of the Euclidean system beyond its intuitive givens,
then we are capable of impact developing different models from a base Euclidean systems. What we know that the Euclidean system, at least in its traditional framework, what you might call to be a one axiomatic theoretical systems and one model group, that they stand in one-to-one relation to one another. Hilbert instead argues through the moving toward an abstract axiomatic system, couch, instead, theoretic terms, you can multiply this, that even though you have one axiomatic system, you can nevertheless have different mathematical models out of the systems
that can or might be even incognizable, might be completely different from one another. And when you say hierarchies, you mean hierarchies in terms of fundamentality? More in the sense of the relations between what you might call to be your intuitive definitions, your common notions, your postulates, and your theorems. That they should not be bunched together according to the same intuitive principles. The intuitive principles only hold for your very base axiomatic vocabularies, namely,
for example, what a line is, what a point, so on and so forth. There is no reason to extend these intuitive relations between your base definitions to higher levels of constructions namely positive a theorem so on and so forth looks like Maria types a question you want to say that Maria allowed oh I just I guess I was googled Hilbert Maria always asked difficult questions by the way just yeah I'm interested in like how like spatially because like how you would think about this like spatially
because Euclid system is like you know you know you're talking about how it's kind of like diagrammatic reasoning and is there like a diagrammatic and I'm just throwing this out there I really don't know very much about like Hilbert or space but like is there like a diagrammatic element or analog to Hilbert I think the relation between the two really requires some heavy heavy mathematical elaboration it's not really it's not even let me put that way in words of space at least from a philosophical mathematical perspective has nothing to do whatsoever with the Euclidean system. Nevertheless, what is really has some relation with the
Euclidean systems, it is Hilbert's understanding of mathematics, namely the role of axiomatic systems. He's a school of formalization, in the sense that what is really mathematics for Euclide is that you can have an intuitive unity at the level of axioms and this is already in the case in case in terms of Euclide systems but the difference is that whereas in Euclide if you have an intuitive unity you only have one model and that's what you might call the conventional e-system Euclidean geometry plane and and the stuff that we were
talking about last session that you know if you have basically an unwrapped ever wrapped two-dimensional plane in which for example two circles that intersect the point will will be on the same play so this is a model that corresponds with the intuitive unity that stands between your axial actions or based definitions whereas Hilbert's tries to show that if you stratify or understand this Euclidean basically philosophy of mathematics across
different levels then you can derive in fact multiple if not infinite models from the same unitary relations that stand between your axiomatic systems that even though what you might call to be the relation between your base definitions your intuitive axioms are a stable you still can develop mathematical models or mathematical spaces out of these disabled relations that can be fundamentally different to one another but this can only be done if you see the relation between intuitive axioms and your proofs namely
constructions no longer in terms of intuitive principles but in terms of pure set theoretic principles that are couched in set theoretic terms It's the idea that you see it can also has and I think as I mentioned I think Hilbert is quite Kentian in this sense You might say that Euclidean systems demarcate the limits of imagination and What is really imagination in Kent? It is a representational relation between what you might call to be a singular representation and an item in the
environment an object it's a one-to-one correspondence namely an image this is idea of image in Kent a singular representation of an object it's a one one-to-one correspondence but if you subsume namely reappropriate or this one-to-one parochial representational correspondence under the category of pure concepts of understanding these are no longer images of concepts pure concepts, then you are capable of reinterpreting this image in different guises.
Imagine like the duck rabbit. The duck rabbit is only a duck rabbit precisely because you are subsuming an image under different concepts and you see it differently. A concept of a rabbit and a concept of a duck. because you have already stepped outside of the rudimentary relation between the image that you perceive and the represented object. You have moved toward the realm of pure concepts that you can see differently. The same thing here with the Euclid, with Helbert's understanding of Euclid's system,
that even though the intuitive import of Euclidean systems allows one-to-one correspondence between the system, the constructive system as such and its definitions, well, once you diversify this relation by way of basically inscribing it in logical set theoretic terms, then you are capable of extracting different models, different conceptual mathematical systems out of one basically intuitively held axiomatic system.
Cool. Does anyone else have any questions before we move on to? No? OK. So let me put the iPad on, hopefully we have a better connection today. Yeah, the connection seems a lot better today. Okay, superb.
who talks about this implicit connection between mechanics as embodied in mechanical engineering made invention of machines and mathematical structures as embodied at that time in analytic geometry is Archimedes this is the so called method the D method namely making explicit their relationship in mechanics and mathematics is it possible if we are encountering a mathematical model to a
step outside of the mathematical structure enter the realm of physics particularly mechanics and search for a solution to that mathematical problem in in a machine. Can a machine, a mechanical machine to be precise, solve our mathematical problem? This is really the so-called Archimedean method, which is based on this assumption that Archimedes' approach of applying mechanical thought devices to geometric problems can reviewed in a more general way than he demonstrated in his famous book The
Method. In this view a mechanical analog is constructed that can be reasoned about more intuitively than the abstract reasoning required to solve the original problem in mathematical terms. In constructing this mechanical analog a mechanical surrogate must be chosen that relates to the original problem through a binding principle the binding principle asserts a set of truths about the behavior of the mechanical surrogate namely the machine that connect the surrogate to the original problem as a consequence we can solve the mechanical
analog as a way of arriving at the solution to our original geometric mathematical problem this mechanical reasoning does not need to replace geometric reasoning to be effective but rather can complement it and offer new conceptual insights and really this is this opening up this new relationship between mechanics and mathematics, between machines and mathematical abstract reasoning is something that is fundamentally predominant in Middle Ages. That virtually the most important and significant contributions to mathematics and natural sciences
have been done through the middle ages not by abiding in the realm of mathematics or in the realm of physics but by creating a back and forth oscillation between physical embodiment physical what you might call behaviors and basically mathematical assumptions and reasoning.
And this form of thought procedures going back forth between mathematics and physics is embodied throughout the Middle Ages, if not from early antiquity onwards, in the relationship between the mathematician and the engineer, mathematical reasoning and mechanical machines. I do not want to go to details about how this fertile relationship between mathematical reasoning and mechanical machines ultimately led to what you might call
to be computer science at this point. But I want to focus precisely on the nature of this going back and forth, this oscillation between machines mechanical surrogates and abstract mathematical principles. So I would like to make a diagram here and that kind of capture the spirit of this oscillation between geometric reasoning and what you might call to be mechanical between the mechanical analog and the abstract mathematical
model that allowed so many scholastic medieval philosophers solve some of the most important problems in physics and mathematics of their own era, but also most importantly, particularly as I will talk about Oreen, they allowed some of these scholastic philosophers to pave the road for a system of reasoning that paved the road for what you might call to be the advent of modern natural sciences, Copernican, Keplerian, and Newtonian revolutions.
So the diagram for the relation between geometrical reasoning and mechanical reasoning can be made something like this. You have two sides. want to get your solutions this is math this is mechanical or the machine this is equivalency and you have your geometric problem
you have your mechanical I don't like problem there is a relationship between your problems also can be interpreted in terms of equivalency between the mechanical analog and the geometrical problem this is the mechanical interpretation and this is the interpretation
for you to get your basically the equivalence relation between your solutions you have here mechanical analog in order to solve it you need to embark on mechanical reasoning to embark on solving your geometrical problem you partake in geometric reasoning and this is really
an overall model of how Archimedes but so as mathematicians and engineers of later liquidity and Middle Ages understood the relation between mathematical problems particularly geometrical problems and mechanical engineering problems so you have a geometrical problem you interpret this on a quote on it you interpret this according to a mechanical analog you simply model your problem on a mechanical analog according to a
principle and then once you have that you can go back and forth to refine the the mechanical the relation between the mechanical analog and the geometric problem once you have done that you work on this side of your diagram once you refine your mechanical analog to make sure that it really does capture your geometric problem then you go to solving your mechanical analog problem by way of mechanical reasoning once you arrive at
a solution then you are capable of transposing the solution this make the solution to your mechanical problem as a potential solution to your mathematical geometric problem. The same also holds if you had a mechanical problem that you wanted to answer, you would step outside of it, model it on geometric problem by way of this equivalence in relations of interpretations and reinterpretations you do once you have a geometric problem that truly corresponds to your mechanical problems and by way of refining its
geometric problem then you are capable of using geometric reasoning arriving at a geometric solution using the same equivalent simulations and positing it as a potential solution to your mechanical problem so this is an overall module of the correspondence between mathematics and engineering that has been extensively used since the time of Archimedes this is in fact really the elaboration of the so-called Archimedean method before I move forward any question okay so again the original geometric problem is reinterpreted using a
mechanical analog. The analog is related to the original problem through a binding mechanical principle. Using mechanical reasoning we can then solve the analog to arrive at the solution to our original problems. This connected Archimedes mechanical ideas with his geometric ideas. When considering for example the volume of a sphere we can come up with a way of so let's let's take a step
volume what you would call to be cylinder volume of this fear now this is a geometric problem how are we going to make explicit or simply elaborate about the relationship between the volume of a sphere and volume of cylinder how can we solve this geometric problem using a mechanical machine hence by mechanical machine I do not mean it
some source of like a clock or something like that at this point what we are talking about are just simple combinatorial machines a lever and inclined plane an axle wheel so on so forth and here again the hint would be kind of a mechanical machine that we want to use in order to solve this problem would be a lever an Archimedean simple machine how can we use a leader a mechanical machine to solve this problem elaborating relation between volume of a sphere and volume of a cylinder any any idea
Yes, it is quite in line with ampliative, explicative, analytic, synthetic distinction, this whole idea of mechanical engineering and mathematics. yes so you like that
So if you put a sphere inside the cylinder, then how much bigger is the cylinder than sphere. Another way of formulating the relation between the volume of a cylinder and the sphere that it envelopes. It doesn't matter if you're coming with the right solution, but as long
as you that would be fantastic if you can just talk about possible solutions to this problem I have no idea okay so it goes like this to answer this question we must think
about Archimedes' principle of leader, which is essentially about the idea of a balance. And it could compare the weight of two objects. If one object was twice as heavy as another, it would be balanced at half the distance to the fulcrum as the other object. using a balance with adjustable lever arms we can experiment for example with wooden blocks of different shapes and sizes and adjust the lever arms to get them to balance
now we know that objects with shorter lever arms were heavier by that same amount by making carefully measured for example clay models of a sphere and cylinder and by adjusting the lever arms we can guess that a cylinder is one and a half times as bigger as the sphere that sits inside of it we were we can uh well of course in experimenting with this mechanical model we need to make sure that we are using materials that have the same density they're the same in order for it to work
so it goes something like this so imagine we have we have this this is our basically problem the sphere within the cylinder we make a lever according to their volumes you since we know for example the volume of a cylinder and volumes of a sphere then we have to in order
to make a balance we have to move this full chrome in a way that you have here an l and you have here three two l l is the length of your lever the problem goes like this that a volume of a sphere times 3 L equals to V of a cylinder times L which then goes to volume of this cylinder
sorry with regard to a volume of the sphere is 3 2 all you need about this is to understand what a volume of a sphere is and what volume of cylinder is then thinking even in imagination how can you balance a lever that has a cylinder on one of its sides one of its palms and a sphere on another the important thing is that in order for this
thought experiment work you need to give them both the cylinder and your sphere substances with the same density the material should be the same once you do that then you are capable of basically moving the fulcrum of the lever in the actual experience or in thought in a way that it corresponds with the volume and the weight of your sphere and cylinder on the two poles of the lever once you
do that you make a balance then you are capable of articulating or elaborating the relation between the cylinder and the sphere that it envelops. There are so many examples of these basically geometrical reasonings done by actual building of mechanical analogs or done in thought experiments simply by referring or delegating your problem to a mechanical engineering problem done in late scholastic and medieval eras.
A very, very important example of this is Orym's invention of formula for acceleration later on becomes a diagram for classical perspective so when you look into you know history books and art books you always see that's invention of classical perspective in painting was due to for example the
interventions of artists of medieval artists but that is absolutely not true before artists actually came up with a diagram the actual diagram of classical perspective it had already been proposed by a scholastic philosophers particularly already um he had already and the interesting really the interesting point of this that there is a connection between combining simple archimedean machines formula for mean speed formula for acceleration and classical perspective So you have a dynasty of problems, of geometric and mechanical problems that are put together,
allow you to transcend to a higher problem that subsumes, or sublite in a Hegelian sense, your current problems into a higher order geometric problem that consists of both mechanical problems and re-eventure geometric problems. By virtue of that you are capable of building higher order mechanical and geometric properties. Classical perspective is one of this. Classical perspective is not pure geometry. It also has, once you start to decompose it, you see that it is really a diagram of uniform acceleration which can also be seen as a frictionless acceleration on an inclined plane.
Incline plane, imagine if you inclined plane, the diagram of geometry, the diagram of an inclined plane and you had manipulated geometrically the way that you could expand and turn to perspective operator then that is basically your diagram of graphical representation of a 3D version of your classical perspective diagram. It's simply an inclined plane that has been basically, whose geometric and physical qualities and the physical characteristics have been sublated into an optical problem so I'm going to talk
about this about basically your how can you get a diagram of classical perspective something like this from geometric problems and physical problems that have nothing to do from you know superficial stance have nothing to do with an optical problem about
perspectivality before then it's important to talk a little bit about who or in videos and what exactly this project was any any question before I move forward Okay. So let me turn off this.
Okay, can you see me? good so or in was a student of John Berrigan one of the most famous scholars in Burton College of Oxford ring was a famous French archbishop and theology economist and natural philosopher in villages who was the head of a school of Paris and he did make significant contributions to economy to banking to
engineering, natural philosophy, theology, and the rise of what you might call to be modern philosophy of knowledge that was completely divorced from the kind of rudimentary relations between science, theology, and magic of Middle Ages. It's also really, if you are interested, you should definitely read his book on money, which is, you know, it's really the first book that starts to elaborate about the abstract powers of monetization and money as an abstract entity.
And it is actually quite astonishing how much his book on money is also connected. His book on forms, abstraction, mathematics, and kind of like his natural philosophical a stance toward physics and mathematics in Middle Ages. what is what we need to know is that orange contribution are by no means
a radical in the sense that he did not simply come up with some of these fundamental insights all by himself is simply integrated them into revolutionary theses. He was like any other, you know, medieval scholastic philosopher, was working in the context of logics, mathematics, theology, and national philosophy of his time. And he was extremely influenced, as I said, by the Mertonist School of Oxford, the so-called
Oxford calculator but so as medieval Arab or Muslim philosophers particularly Abicina and the reason that I'm particularly emphasizing Abicina's basically philosophy in relation to Aurine is that so many historians believe that Abyssinah's commentary on the relation between mathematics and physics is essentially the fundament of the entire process of what you might call to be empirical sciences that he simply articulated in very basic terms a relation between physics and mathematics that paved the road for
or Mathematization of Nature, the so-called what you might call to be the advent of Enlightenment Sciences. that one most important observations about Avicina's relation between physics and mathematics is the relation between what you might call to be quantities and qualities couched in vocabulary of a
scholastic philosophy this can be thought in terms of the relation between the triad of the so-called dimensional sorry quantities dimensional this quantities virtualis and for moral corporalis so what are these you know three entities that constitute the triad that make up the relation or elaborate the relation between physics and mathematics so quantities dimensional is very roughly
speaking is about the kind of quantities namely degrees of intelligibility when we are talking about quantity at this point in the scholastic philosophy of Middle Ages we are not talking about literally about what we mean by these days but it has also a conceptual input quantity in middle ages is about degrees of intelligibility of nature so there is a direct connection between the development of the system of quantification in this time and the metaphysics of intelligibility of those
which exist now so quantities dimensional is tries to or basically pertains to the distinguishing features of a body in a space in terms of length width and depth this is called quantities dimensionalis that allows a body to receive this or that form of determination quantities virtualis is a
basically different from fundamentally different from all the human who is connected to the quantities dimensional it's different from quantities dimensional so what you might call to be the extensible properties of the body quantities virtualis are about the undetermined or indetermined qualities of a body in itself what you might call to be the subjective properties that partakes in one or more qualities like heat and cold and these quantities virtualis can be thought in terms of for example it's getting cold
this this bar of irons getting cold this bottom irons getting hard or for example this projectiles is moving fast this projectile is moving the slope velocity or heat in this sense the quality in which subject namely a body partakes and this act of partaking in a specific quality can be also thought in terms of duration of time along this duration the body X partakes in such and
such qualities in such and such degrees so this is you can see that quantities virtual is is connected with the idea of intensibles namely intensity in in the vocabulary of philosophy whereas quantitis dimensionalis are about extensible properties namely extensional properties of a body or subject so in addition to quantitis virtualis and quantitis dimensionalis we also have formura corporalis
So formula corporalis, you can think about this as the relation between intensive qualities and extensive properties, how intensive qualities give some characteristics to extensive properties and how extensive properties, namely quantities dimensionalis, elaborate certain intensive characteristics namely quantities uh virtualis this is what you might call to be corporal or corporeal form of a body corpora for a form of performance so everything in the vocabulary of middle ages and particularly for abecina abecina is the first one
what we actually come up with this triadic way of looking at bodies or subjects. Everything in Middle Ages is laid out in terms of the relations between these three compartments or three essences of a body or subject. The relation between the intensive, the extensive, and the corporeal. And the corporeal is simply what you might call to be the, is afforded by the individuating power
of intensive and extensive properties. So we essentially have, And the whole idea is that this relational characteristics for any given body are essentially telling us about how we can render this or that body intelligible. So the purpose of laying or characterizing a body in terms of quantities dimensional is quantities virtualis and former corporalis is to come up with a system that allow us to methodically start to think about
what bodies are and how they behave intelligibly in terms of intelligible properties so we see that that's basically from this perspective people like abyssinore the ream lay out the foundation of modern system of knowledge is based on elaborating intelligible properties of an object or an item and as we see we already have a
properties and we have some material characteristics like mass we're always constant we can see in this formula something like the virtual characteristics that that allow that basically start to fold and envelope variation of extensible properties of body in the space in terms of characteristics of pure material properties
as hinges so you might say that something like virtual characteristics hinge extensible properties and pure intensible properties they allow us to articulate variations in terms of invariances or explicate an invariant property like something like a constancy like a mask in terms of variations like here in this formula like volume so for any articulation of anything intelligible
in the world, any object, we essentially need not only the extensible properties and the intensible properties, but something that mediates the relation between the two, the so-called hinge that render this system of intelligibility possible. hinge are called virtualis or quantitas virtualis or virtual quantities virtual properties these virtual properties you might think about them in terms of how some purely
intensible properties are being distributed in space. And how if you have for example the extensible, namely distributed properties of an object like a projectile in space, then how can you reconstruct these extended characteristics, the spatial characteristics, namely the quantitas dimensional is back into those pure intensible properties how can you make intelligible the relation between purely intensible and purely extensible you cannot do that unless you have something like dimensional is sorry
quantitas virtualis that you can think about them as hinges between intensible extensible properties so before I move forward and elaborate how or in starts to elaborate this and turn this into a very inventive way of solving the problem of acceleration is there any questions so quantities dimension analysis extensities. Extensities, like completely a Spinozian properties like of a body in a space. And intensities is quantitas virtualis, is
that what you said? They are, in medieval philosophy, they are associated mostly with pure intensities. They are what you might call to be are ways or relations by which a subject partakes or participates in its qualities. What does that mean? Imagine a bar of iron is a subject the bar of iron is a subject you heat it up with flame how does the subject participate in the quality of being hot now you see as you
heat up this bar of iron something happens to it it expands over time the expansion of the bar of iron are its extensible properties it's completely intensible undetermined generic properties are what called to be our pure intensities the way that in by which the ways by which it partakes in the quality of being hot so as to extend such and such in a space are quantities virtualis quantities virtualis are not given to us but the
difference what is given to us are extensible properties because we can see how a body expands in space the pure intensities for example something like mass in our formula of m equals to dv are not are undetermined we absolutely don't know what the fuck they are but we can create between what is given to us and what is completely undetermined by creating a hinge a system that starts to elaborate how something that is purely intensible can be correlated with something that is purely extensive these are quantities virtuals you might call them are basically ways of making
an object intelligible in a space by elaborating how forces or kinematics or dynamics are being unfolded and laid out in terms of a spatial extensible properties Sounds good. Great for me. So, by the way, should we have a break? Like five minutes and then come back?
That sounds fine. Are people okay with that? If you don't speak up, I'll assume that's okay. That would be great. Okay, so far. See you in five minutes. Sounds good. Thank you.
Daniel what do you mean by modal status of axioms in reverse systems? The level of logical relations with axioms, what you might call to be the micro-theoretical
relations, at that level the relations are usually necessary in order for you to have the possibility at the level of macro logical theory or macro theory, namely the space of model constructions they can be they can be treated or intuitive I mean logical relations
between intuitive axioms but they can also be fundamentally abstract axioms again completely put in terms of informal non-formal set theoretic terms I mean the point is that Euclid axiom is intuitive definitions even though it has intuitive definitions once you see the relation between the axioms completely in logical set theoretic terms you can derive different models from your intuitive axioms but not all axiomatic systems have intuitive axioms like euclid some have and this is a
Sekulers and also Lawrence Ponton's point of view that axioms can be fundamentally abstract laid out in set theoretic predicate logic terms So it should be a start? Sounds good. Yeah, it's been a little more than five minutes. So. Okay. So I think I've already noticed this triptych of characteristics
pertaining to an object, material characteristics, virtual characteristics, and dimensional characteristics, corresponding to pure intensity, intensive characteristics, and extensive characteristics, can allow us to think about how we can render any quality whatsoever that we
see in the world intelligible in terms of degrees of intelligibility this is the project of quantification quantification project of measurement is the project of articulating degrees of intelligibility for any given quality no matter how intensive it might be. Or how convoluted it might be, like trepidation, heat and cold, like motion in motion. You know, motion from a scholastic perspective is a quality, from our Italian metaphysical sense.
But we just, sometimes we don't have just one motion for a projectile, we have sometimes motion in motion. Isn't it the whole idea of non-uniform acceleration, motion in motion? So if this is the kind of convoluted quality that an object has, how can we render this intelligent? How can we quantify it? This is the project that essentially lays the groundwork for the advent of the Copernican Revolution. And the person who takes this very seriously, makes the most significant contributions, is Nico Morini. And in fact, majority of historians, as I said,
think that Galileo and Copernicus, equations of motion with regard to celestial bodies, were completely derivative with regard to Reims diagrams, Reims system, graphical, geometrical system of studying the motion and velocity of bodies. In a sense, Copernican revolution wasn't a radical breakthrough but simply a modification
between quantity and quality in terms of Avicina's triptych of pure intensity, intensible and extensible properties that I mentioned. We know that velocity is something like delta x divided by over delta t. Change of distance with regard to change of time. Now, from modern perspective.
we might see this to be representing a graph in which you have you can build a vertical line representing distance L or X and a horizontal line representing a T and then saying that the line did that go online this one the hinge between your vertical and horizontal is L equals to VT and then
basically every parts every instance or every moment of your motion for a given projectile simply will be articulated by a bit of y axis and a bit of x axis a bit of distance and a bit of time this is you know from a modern perspective of a graph this is completely fine but Oren did not go for this solution and in fact as I will argue or even solution was much more non-trivial and advanced
that's then the modern graphical representation simply by virtue of understanding of what it means for this hinge for this graph to connect the vertical and horizontal the extensible properties and the intensible qualities expressed in terms of duration time by creating device or a diagram that could fold durable or durational intensive properties and unfold them in terms of extensible properties or the
other way around and below pink extensible properties back to the purely intensive intensive properties or unfolding intensive properties in terms of extensible properties that we can measure everything quantify this hinge this diagrammatic hinge that never that didn't see the idea of intelligibility of a body in a space no longer in terms of simply coordinates of y-axis and x-axis
intensible purely intensible and purely accessible properties but only as a breath or amplum and hence the word amplitude and clone of degrees that are capable of folding and unfolding going back and forth degree by degree between extensible and intensible properties is the very idea of quantification and Reims had a fundamental had a significant emphasis in how this idea of
ample the hinge between extensible and intensible can be expressed in terms of diagrammatic representations laid out in analytic geometric terms that can be traced and elaborated by way of Euclid's system now let's see how this works First, a rudimentary example that I already mentioned.
Imagine you have a bar of wire which you are heating. so the quality that the subjects called a bar of iron is participating in partaking in is heat or hotness how are you going to render the idea of hot or heat or being hot for this bar of iron intelligible this is essentially the idea of quantifying the degree of heat
within a single subject here the bar of iron so understand this that you are heating this bar of iron the bar of iron is going to expand what is the expansion of the bar of iron. Extensity. Yes. Oops, sorry. So, some.
so first at time t1 it expands this amount t1 let's say to t2 t2 to t3 it expands this amount and 3 3 to t 4 it expands this amount expansion of viral viral are the
extensible properties of the body that can be laid out in terms of quantities dimensionalis, length, depth and width. What you might call to be the formal properties in today's physics sense is the spatial properties. Now if we have this if you have so this bar of iron is participating in equality which is heat in hot by virtue of participating in this quality it exhibits some
extensional properties namely expansion in the space according to some duration or time now if we have this expansion namely the bar of iron expanding from time to time how can we say or elaborate a precise degree by which this bar of iron has participated in its own quality in being hot the project of
quantification come up with a diagram that can accurately represent the not only the expansion of the bar of iron but its participation in the quality of of being hot any idea so the example that our reason are even did not use is the time is the x-axis and expansion is the y-axis yes yes so basically the y-axis what you might call
to be the intensible at a specific time the intensity at a specific time the quality of a specific time this is what or in calls latitude of four forms in medieval vocabulary are simply synonymous to qualities forms or receive form in medieval text is simply quality so orange called these vertical axis corresponding to the y-axis in modern sense latitudes of forms namely the intensity of equality at a given time he calls the x-axis a longitude longitude
are what you might call to be the subjects participation the the subject duration of participating in its quality now from this perspective X axis does not represent extensible properties but purely intensible properties participation in its own quality like a bar of iron so we have something like
this a time t1 to t2 the bar of iron grows this amount at t2 to t3 the bar of of iron grows at this amount, T2 to T3. And finally, oops. the bar of iron grows this amount.
If you see the time vector or the durational vector which corresponds to your x-axis is simply a register of the durational characteristics of the subject participating in its own quality pure intention intentional or intensible properties basically the so this is the longitude this is the latitude the latitudes are the one that you can say
are more in tandem with what you might call to be extensible properties but that's not purely correct because latitudes so you have this in the in the modern version is something like this so you have a line and you have x y axis and then any point of your line has a bit of x axis and a bit of y axis but here is a different things your latitudes are not being diagram the represented simply in terms of lines but in terms of areas
namely how much extensive properties unfold the irrational properties as you see the area here of each of these rectangles represents the amount or the degree of the quality expressed in durational properties this is the idea that's in a scholastic philosophy they call it ample in modern terms and call it amplitude the breath the width So the amplitudes, you might see them not simply in terms of extensible, intensible
properties alone, but in terms of how they envelope, how extensive properties develop intensible properties and how intensible properties envelope extensible properties this is the idea of amplitude the breath the idea of a hinge breath is essentially just the amount of change that the subject undergoes when it's exposed to the flame or the variable yes yes which can be if you represent it in terms of area rather a
precise coordinate that has some bits of x and x and y axis if you represent it as an area then you can represent it in terms of degrees of folding and unfolding between intensible and extensible exactly like the formula M, D, V. D is density. V is volumetric extensible properties, completely variational. M is invariant constant. d is something that hinges an ontologically precedence
m an ontologically consequent variational term called v volumetric variations It does not simply like the modern graph is something that has a bit of this, a bit of that. But you can think about this D, this hinge, which here in Orym's diagram is expressed in terms of the area of rectangle, as something that captures how much degree of the variation can be attached to something
of invariance and what it means for this constant property or quality of form or invariance at this time, or this duration of time, to be unpacked, decompressed, or unfolded in terms of such and such extensible properties. Any questions?
Can you think of an example that's not using area or is area? Oh, area is absolutely necessary for you. Area is basically the whole idea of elaborating the relation between pure intensible and pure extensible, not in terms of some pure determination, but in terms of unfolding, in terms of degrees of intelligibility. of intelligibility is the very idea of quantification measurements. It is not arbitrary, it's a matter of determination. And determination is not something that is given, but it's a matter of procedure. But could we use something like, instead of using area, could you use weight or mass?
Area is simply a graphical representation of these things. I see, I see, I see. Okay, okay. Yeah, yeah. So he wants to come up with a universal graphical representation of elaborating this hinge, density, or for example, velocity, acceleration, so on and so forth. a hinge between time and space between durational properties and purely extensible properties and you see the relation here that I mentioned that
construction of new spatial invariances has something to do with our idea of what is intelligible in the world first few sessions and we're talking about gestures and construction of the spatial invariances and this is it the construction of a spatial invariances that are capable of connecting between the rational properties of a matter and extensional properties of body are absolutely connected to what we know as
making something intelligent without this hinge without these intermediating spatial constructs here represented by an an area in Orien's diagram, we couldn't have something like an intelligible object. So, let's...
Something that just occurred to me is like maybe like the intensible qualities and like the extensible qualities are like the relationship is like the relationship between the first and the second line in the Plato metaphor. it's all about how intensive quali- how we can comprehend intensive qualities through the the what structures of intelligibility we need in order to constitute at the second line as a understood constituted objectivity oh no he left yeah i think we just lost him um we'll just wait for him to come back Can you expand on your point, Christian?
I was just thinking, I mean, it seems to me, or like not necessarily, I don't know, because it's like, it's like the universal conditions to express types of qualities as extensible. So, I mean, I'm just trying to put it into terms of the Plato, the four line metaphor in Plato. Right. Because, like, I mean, on one hand, like, it seems like the intensive side is definitely... Sorry, I got kicked out. Okay. Oh, basically, I was just asking, like, how this fits in with the four line metaphor in
Plato. I mean it seems to me like the intensible side would be like the first line with the indeterminate qualities but in order for them to be fully expressed you'd need some structure of intelligibility to make the constitute the object to be as such. well the whole idea is that these so-called hinges quantities virtual it's are really the whole idea of mathematics method in the divided line they refer precisely not to the epistemological but the metaphysical status of mathematical or analytic idealities in the divided line namely the third
segment being posited between pure idealities and so-called objects in nature allow you to bridge between multiplicities of objects in the world variations and extensible properties and in variances that are given by the metaphysical status of pure ideal realities forms in Plato have reality and that's why so many people believe that Plato's ontology the two-level ontology objects have their own reality ontology but that reality is ultimately
dependent on the reality of forms Which are one, which don't have any multiplicity. So how much time do we have, by the way? So we're at 3.20 right now, so we have about 10 minutes. Okay, so just, I will, so I think that we will go next session and I will talk more about the details about diagrams. Well very briefly to wrap up today's discussion in terms of what we were talking about.
so you you you you you you you you you you you what is interesting in Reims formulation of Abyssinus principle of triadic entities of any intelligible system is that precisely because the
mediating term named the quad is virtualis is graphically represented in terms of an amplitude an area that envelopes some degrees of variations it already suggests or be speaks of this fact that precisely because graphical representations or the third medium that the intermediating medium the so-called quantum spiritualist and phones variational degrees depending on how we
can graphically represent or geometrically pose this amplitude or this intermediate term we can come up with new intelligible properties with regard to an object so within the metaphysical status of Orym's diagram we can already anticipate that's it for example if we graphically represent quantities virtualis as an area that envelops some variations we can elaborate
or we can demonstrate come up with some intellectual properties about a body that participates in equality and exhibit such and such extensive qualities but it also means that precisely because the graphical representation this bit the the environment of variations could be thought not in terms of simple Euclidean area, but in terms of other geometrical entities. Which means that if we have, and this is more in line with the metaphysical epistemological
status of Abyssinian's and Orem's diagrams, if we had different geometrical entities that could express this envelopment of degrees in a different geometric form rather than simply a euclidean area then we could make explicit further intelligible properties about an object and this is exactly what you see in post-Galini and Copernican relation between physics and mathematics where mathematical models particularly geometrical models give us more intelligible properties about the
behavior the extensible behavior of an object in a space and time another interesting point about Euclid, sorry, Orem's Diachromes is that if you have already noticed, We have some pure intensity, we have some pure extensity, and we have some amplitude,
some breadth, some degrees of folding and unfolding between them. This plasticity, this intermediating plasticity that is between elasticity and pure extension, intensive elasticity and pure extension can be thought also dynamically in terms of what it means what is exactly the mechanical function of this plasticity how can we diagramatically represented how can we think about this plastic plasticity or amplitude that is captured but there's capable of
intermediating between intense pure intensible properties and pure extensible properties but without being reduced to either one of them uh as i will talk about in the next session this is essentially um the the idea behind or even diagrams of a mean speed and uniform acceleration in the sense that what we are dealing here you can think about the amplitude namely these intermediating
entity in terms of some sort of process of contraction and the contraction something that contracts extensive properties to intensive properties and contract intensive properties to access properties a good diagram of this but what I mean and of course I will talk about how this is related to the formula of media speed and acceleration will be something like this that's you have
you Can you see the whiteboard Yeah, okay, so if you remember you had L and you had T so you had X and Y axis One more vertical and what what's horizontal the amplitude is not something like this the modern graphical representation the amplitude is something like this think about this in terms of a machine a
mechanical engineering problem if you were going this is degree one of velocity or intensity and this is a degree zero of intensity between one and zero there is a continuum of degrees or variations exhibited by the so-called ample or amplitude or the hinge quantities virtualis now quantities virtualness you can think about it as this series of hinges and joints that you can compress it or decompress it like if you have one of these joints you
I'm sure that you have seen this, these hangers that fold and unfold. And this is exactly like that. The ample represents a continuum of degrees that you can, according to the principle of how you articulate the relation between your extensible properties, and extensible properties unfold it and fold it. between one and zero there is a infinite continuum of degrees so this is not like a seismograph where it no yes this and folding and unfolding degrees as I mentioned or it
represents them by way of area area formula in the Euclidean system what is area exactly area is L of a rectangle L and its length and width how they can be connected to one another can be expressed as the area you can express this how L and T 1 and 0 length and width can be connected in different
schemas rather than just area depending on the kind of geometric system or mathematical structure you are using so you can essentially have different classes of intelligibility between an object participating in its qualities and its extensible extensional properties its behavior in the space and time so I'm sorry I don't
understand the zigzag type graph yet what it's measuring or how it's measuring this the zigzag measuring is simply what you it's simply what you might call to be the mechanical diagrammatic understanding of how one and zero length and width or for example length and time distance and time are going to be translated to one another basically how they can be translated to another by creating different continuance amplitudes of degrees and how can you
really put how can you express these amplitudes can be numerous but simply all you need is a mechanism for compression and decompression here we They're just coming with a rudimentary idea of compression and decompression. It's like the idea of a joint system in mechanical engineering. You create joints, then if you compress it, it will become a vertical line. If you decompress it, it becomes a horizontal line. Okay, yeah, I understand that now. We will talk about this how or even really implements this to come up with this idea of meanest speed and the formula
for uniform acceleration, allowing him to understand what is exactly the nature of motion in motion. Because in Aristotelian metaphysics, you don't have motion in motion, you only have a motion. In fact, motion in motion is a contradictory term, but you can have motion in motion. This is the very idea of acceleration. If you have a motion in motion, then you are capable of talking about various rates of motion among celestial bodies that cannot be convincing. And hence, you can think about cosmos differently. And that's why basically Auriem is extremely important
from a historical perspective in giving rise to the Copernican and Keplerian revolution. Again, as I said, motions are quality. Velocity is a quality from a scholastic perspective. a body that participates in motion is a is a subject that participate in a specific quality the quality is not given to us we need to render it intelligible namely understand or elaborate the relation between the
position and time of a projectile at time x1 t1 and its intensible properties but also capable of not simply talking about these positions in discrete time and a space but in regard to one another as a configurations and this is the idea of ample ample on you see in in all the diagrams in in modern diagrams we had this that for example time x1 time x2 t2 you see we have this as disjointed instances of projectile moving at uniform acceleration that can be at each
instance can be expressed by a bit of time exact it's x-axis and a bit of y axis but what ORIMS manages to do is capable of configuring and showing the relations between each of these instances of the projectile at time x1 t1 with regard to time x2 t2 with regard to time x3 t3 ad infinitum so he is capable of intuiting the motion of a body in a single coherent intuition not as
disjointed degrees but simply every motion with regard to the previous motion precisely because you have this comparative and a logical system now then you are capable of also thinking not about uniform motion but also non uniform what or it calls deformally uniform motion non uniform acceleration motion of projectile that's at time t3 in terms of time and position is fundamentally different from its previous motion at time t2 and x2 so you
have done uniform acceleration so you have the idea of motion motion precisely because you were capable of moving from this disjointed view of qualities the relation between intensible and extensible to the amplitude you know view of how extensible intensibles are related not disjointedly but in terms of degrees of variations and then you see that basically this ultimately this understanding if you
don't have this understanding you never have anything like the idea of acceleration acceleration would be impossible for you to think about because acceleration essentially requires degrees of variation between intensible and and extensible degrees that can be quite also not uniform and if you didn't have accelerations you couldn't have equations for the motion of celestial bodies if you didn't have those equations you couldn't have anything something like copernican revolution or keplerian revolution or newtonian revolution
um so is the rendering intelligible about the dimension of an object like a single sort of variable or is it claiming like a being of the object itself the intelligibility is not as I mentioned is not about the extensible versus the intensive or intensive versus extensive it's about how we can bridge the dimensions to qualities quantities to qualities extensible quantities to intensible qualities so so is there a degree of full intelligibility I think from this at least from a metaphysical perspective
the degree of full intelligibility is erroneous precisely because from at least this classic perspective of metaphysics materia matter precisely because of its potentialities can be expressed differently in terms of extensible quantities and if you are and precisely because of this then there are infinite ways of making this matter and its extensive properties intelligible given
the fact that the potentialities of matter at least in scholastic philosophies are infinite. Its extensible, its intelligibilities are also infinite. So from at least from a scholastic perspective there is no exhaustive form of intelligibility. In fact the idea of intelligibility from a scholastic perspective points to the inexhaustibility of what matter is. Remember, in the formula N, D, V, N is the one that has ontological precedents.
D is the intermediating term and V is the variational or extensible term. N is the other side of the formula, precisely this shows this ontological precedence that D no matter how much you build it given the ontological precedence of M there are infinite ways for increasing the intelligibility without ever exhausting the pure intelligibility of M as such.
Any questions? Philip hasn't been silent today. Yeah, I don't have any questions. . You know, I was just watching the up and down variation as the thinking of the sine or cosine wave going from 0 to 1 to 0 to 1. That's my only . Yes. No, it is actually quite interesting to see that even the graphical representation that we're in offers, you can see it in every basically avenue of mathematical physics these days.
It's simply the idea of mathematical physics, the relations between mathematics and physics, mathematical structures and physical properties particularly in terms of motion namely extensible properties the only reason I will talk next session the only reason that's or aims couldn't come up with something like full-fledged Copernican revolution was because he did not have something first of all he most important thing is that he didn't know the implications of his own diagrams. He had built this sophisticated geometric systems
for representing the relation between qualities and quantities, but he didn't know exactly the implications of his own diagrams. One, but also second, is because of his, precisely because you know at this time you don't have anything like differential calculus all he had was analytic geometry namely euclidean system he could not have something like limit functions that allow him to represent these diagrams in terms of curves current modern understanding of a graph hence he couldn't really understand what it means for something to have a form of motion that is deformally deformally deform namely non
uniform acceleration instantaneous rate of change that allow you to think about various motions with regard to one another these are all of course analytic geometry already gives you a very very approximate precise approximate way of understanding what a curve or a graph can be this is as exhaustive method again invented by our activities you can in fact calculate the area underneath a curve by way of rectangles by creating ever smaller rectangles under the curve there's a method of approximation but nevertheless it is not really give you a
way of understanding what the curve itself is that's apt to wait for the invention of modern calculus particularly the introduction of limit function by Weistrosse Okay, questions before we end today's session?
Well, there are all sorts of prefigurations of the calculus, right? The limits of exhaustion by eudoxys and so forth, kind of filling out circles with squares and hexagons and triangles. Yes, yes, yes, absolutely, yes. method of exhaustion is quite actually integration it's a form of it anyway yeah yeah absolutely I will talk about this how the Orems diagrams actually a
method of integration next very very quickly you see it the whole idea is that for example think about our war of iron that for example in each time period it expands such and such amount so you have some x and y axis and you have for example this expansion at this time you have this expansion at this time and then you have this expansion this time this expansion at this time now of course you can create something like curve or of course he didn't have that so he was trying to think about uniform expansion of our iron which you can easily connect
the head of these triangles by a line this line called it the line of summation the line of summation is absolutely the idea of an integral Right, and the other thing I was thinking about, I mean just free associating, is of course we read the dimension of time as a form of the coming, but that particular mode of reading what we were doing with the methods of exhaustion in classical times, and maybe what I'm doing here is channeling a little bit of Spengler's notion of incompatible number worlds or math worlds which I was just glancing at for some reason yesterday. It
can never see what it's doing in classical times as the unfolding of a time dimension, right? yes yeah absolutely yes I mean the idea this idea of becoming I think in fact is a little bit misleading when you know we we start to think about the relation between intensible and extensible even putting modern physics it's very misleading in modern physics essentially it has changed but without the idea of metaphysical becoming or de-rational properties and this is exactly what
every change can be expressed in terms of metricization of the spatial properties right quantum entanglement yeah I mean basically any anything that you have in quantum mechanics in modern physics is really metricization of the space you can all you know forgo with the idea of the irrational time or become I was just gonna ask what why or actually you're still screen sharing if you wanted I'm lying yeah but why is why are you Philip why are you saying that
it's a measurement of become becoming instead of whoops something happened here. That looks crazy. That's better. Oh, no, it's doing it again. OK, go on. What I was just saying is from differentiation and the calculus and its use in differential occasions to measure variations in population over time or so forth, the sort of mainstream of, I guess, thought about what calculus was doing was it was unable to represent becoming or unfolding over time.
I think Reza's point is right, is that once you get to quantum entanglement where you can't really locate or things can be in several places, you can't really talk about a smooth unfolding becoming for each particle. You see, Theodore, it goes like this. What is really, I mean, essentially the basic idea of differential calculus is instantaneous rate of change, namely the rate of change between two rate of changes. Now, for each of these rate of changes, you can lay them down in terms of for example the relation between extensible properties and durational
properties but if you are trying to study the instantaneous rate of change the relation between rates of change then you can factor out the durational properties precisely because they share these durational properties and simply talk about these rates of change in terms of excess extension properties Right, right, I see. I have nothing to add right now. OK. Thanks, everyone. Next session, I will continue with our Orain thing. And then I will go to, after that,
Also I share a couple of texts, old manuscripts. I will talk about the invention of algebra and a study, geometrical study of quadratic equations. Should we read for next week? Well I have put a couple of books already in the Google Drive. One is the, I will add some more. I will try to put Nicola Rehm's manuscript, but I've already added this book called Engines of Imaginations, which is about the advent of mechanical engineering and its relation with geometrical problems in Middle Ages.
OK, great. Great, thanks, Reza. Absolutely. One question. Sure. So I don't know mathematics too well yet, but I know the whole idea of spatial invariances has been really important. And I totally agree with that. I was wondering if, for example, the intuitions that build the intuitive steps necessary to understand non-Euclidean geometry, would they be built in Euclidean geometry? Like what like it seems to me like the whole spatial invariant side is really important so it seems to me if you're
understanding this space without a scalar invariant you would need to have a separate space with a scalar invariant in order to do calculations in such an abnormal space you see I think the answer to this a little bit difficult Euclidean geometry yeah you're for example thinking about Euclidean geometry and geometry it can be you know expressed in terms of you were talking about but in all of the higher order geometries the concept of a space is not an intuitive concept of a space it is in fact what you might call to me
is in so when you have for example any theoretical systems usually have different variations of key elemental concepts some of these elemental concepts ascend into one-to-one relationships with your axiomatic terms this is the case with Euclidean axioms and Euclidean systems but sometimes the concept of a space is not about your axiomatic term but about the coherency the logical coherency and consistency which is abstract between your intuitive definitions and in majority of the higher order geometry is the concept
of a space is not really about the intuitive components on your definitions what about the inferential coherence of how these axioms how these definitions hang together in a logical space that logical space is what defines the concept of a space for rather simply being traceable to you know your intuitive components like Euclidean space so I think from this perspective it's kind of difficult to talk about the concept of a space in higher geometries with regard to the concept of a space in Euclidean geometry in which the concept
not a space is in a one-to-one relationship with the intuitive, perceptual components of your definitions. All right. Yeah, I was just kind of wondering if you would need like, like, Lydian intuitions to bootstrap it, but I guess not. What you have to bootstrap is logical relations. That's exactly what Hilbert's trying to say. that even if you have intuitive, fixed intuitive axioms, by virtue of establishing logical abstract connections, couch in set theoretic terms, between your intuitive axioms, you can create different mathematical models from the same basic axiomatic systems, which might
have their own different concepts of space. Yeah, two comments, Christian. One on your notion of invariance and transformation. There's a whole kind of ladder of geometries of different generalities, even before you get to non-Euclidean geometry, where you're trying to figure out what remains invariant under transformations. and in Euclidean geometry, transformations are things like rotations, translations, or reflections. And those have to be distance preserving in Euclidean geometry, and that's called metric space.
But then you have all these other geometries like affine geometry, projective geometry, and similarity geometry, geometry up to topology where less and less remains invariant. When you finally get to topology, the only thing that remains invariant is the description of a neighborhood that points there into. So you can think of that as either going down from topology all the way to Euclidean space as a very, very particular and very specified form of geometry as opposed to a more general geometry. The other thing is when you're going into non-Euclidean spaces,
or even I think as Reza said, there's a point after you get from three space, you know, two space is the plane, three space is the plane, the space we sort of intuit, when you go to spaces 4, 5 and N, you switch essentially from geometry to algebra because you can't literally visualize these spaces or intuit them. So you have to describe them by means of numbers, differential geometries if you're doing changing curves and stuff.
Thanks very much, Philip. Appreciate that. OK. Thanks, everyone. All right, see you all next week. Absolutely. See you next week. Take care. Have a great day. Bye-bye. Bye-bye.