Hello and welcome to the 10th session of Maths and Ideas. Here's Reza Negresan. Thanks everyone. Okay, so we ended the previous sessions with a little bit of material on the most basic forms of algebraic expressions, quadratic equations. So today I'm going to talk a little bit about, you know, this proto-history of algebra,
developed from antiquity to mid-Renaissance under the guise of arithmetic and analytical geometry. Now so many historians of mathematics, particularly Sabathai Orobu, the Romanian author of mathematics, they believe that we can't really find anything resembling algebra
in before the advent of what you might call modern algebraic expressions expressed by symbols, meaning that essentially the achievements of Babylonians, Egyptians, Chinese, Indians, and you know later on people like Descartes weren't really developments in algebra but simply pushing you know attempts to push arithmetic I mean the elementary at the
arithmetics and geometry to their ultimate conclusions and we should not see these as any form of genuine algebraic developments. So the material that I'm going to present is that I'm definitely going to regardless of this conclusion I am going to present the kind of material that in fact shows that if even if this is really the case nevertheless the conceptual basis of and modern algebra were developed far ahead of its advance.
So a very, you know, brief, since we are not going to, you know, overstep the boundary that we initially demarcated, which was from late antiquity to renaissance i'm not going to talk about what algebra is in details but i'm going to point out a few important points about the rise of algebra and why is that algebra
such a powerful tool and to the extent that in modern mathematics algebra is considered to be the king of all mathematical fields. We talk about this idea that geometry, at least in its Euclidean pre-modern manifestation, is tied to intuition and experience, at least at a level, at some scaffolding level.
You might see algebra as being the first field of mathematics that ejects the experiential that for so many philosophers was playing mathematics. And in that sense algebra from a modern perspective is the reclaiming of the sovereignty of mathematical
fields as distinguished from the kind of sciences that are at some level connected to the objectivity of experiential content. And but why is that that this is the case? Well, there are so many reasons that I mentioned. If I want to really articulate these reasons, I will violate the boundaries that we have been demarcating so far.
But very briefly, and even though this is a very rudimentary and somehow trivial point, nevertheless, it has huge consequences. And this particular reason is the use of symbols, namely a form of mathematical expression in which we are capable of studying magnitudes away from the burden of imagination that has been the driving force behind Euclidean geometry.
Once we are using symbols in mathematics, we are capable of, as Katharina Dutelnou-Weiss says in her book, which is about the logic of formalism, is that we achieve something quite specific in what you might call overarching cognitive sphere and that's
the capacity to rest an expression from a particular meaning or semantic connotation. This is really the power of symbols that ultimately leads the computational revolution, the power formalism that once symbols are being treated as autonomous and all we are dealing with is the relationship between symbols without any implicit or
explicit connection with a particular context or experiential content then we are capable of extracting these relations between symbols and reapply them to different semantic contents. So there is this process that is happening at the deepest level of algebra. And this is the process of formalization, which entails a form of duality between de-semanification
and re-semanification, in the sense that symbols are de-semanified machines, in the sense that they do not have any binding relation with a particular semantic experiential content. So this is the semantified dimension of the symbols. They are purely abstract. But the pure abstraction also means that once we understand what de-sematification is, we
can begin to employ symbols in the service of re-sematification, namely applying them problems to particular contexts which from the first glance from our intuitive glance couldn't accommodate these applications so the semantification and re-semantification resting symbols resting mathematical expressions from particular experiential content or semantic context allows us to reapply
these symbols these abstractions to new contexts and this is really the power of algebra at space another power another another reason that algebra is is important and there is such a high emphasis on algebra in modern mathematics, precisely because algebra is ultimately a science of magnitudes. We have already came across the significance of magnitudes in the field of mathematics
following Aurim. And the whole idea that mathematics becomes, since Aurim, becomes a science of rendering intelligible qualities of nature, or any intensive quality. And so far as quantification, the science of quantitudes and the science of magnitudes are related, if we are capable of forming a new field of mathematics in which the entire emphasis is being put on the discovery of new forms of magnitudes unburdened by our experiential
limits then this is a huge move forward in the project of rendering the structure of the universe intelligible and hence why is that algebraic notations play such as fundamental role in mathematical physics another reason is that so far in the kind of mathematics that we have been looking at, we don't have what you might call
to be an effective calculus of problems by problems here I mean a form of understanding addressing and ultimately resolving unknowables unknowments by virtue of the data that we are given we talked about this idea that's in Euclidean system ultimately Euclidean system is based on data the data that
you are given in fact the name that the word that means they give it so we start with our given data we construct some new conclusions but what if these datas is that in the sense of the givens are limited but also they are constrained by different kinds of unknowns.
In algebraic notation, you might talk about these unknowns in terms of x, y, z, whatever you might call them. So this whole idea that when it comes to the mathematical project working with the data, we need to, and this is the algebraic revolution, that we need to suspend the idea that the givens named the data are always unlimited and we can employ them in whatever way we want in order to reach certain conclusions algebra tries to
treat these givens on a different plane basically creating a universal to the the method is universalist. In order to handle the givens so as to resolve unknown or deal with variables that from a geometrical perspective cannot be solved simply using our given data. So these are some of the reasons that algebra became popular after the Renaissance.
And for a good period of time, algebra completely replaced the sovereignty of geometry. But of course in late 20th century, you know, mathematicians started to question about this kind of a bipolar opposition between algebra and geometry and started to create a form of what you might call to be a compatible, a compatibilist approach between geometry and algebra in which the kind of geometry that we are thinking about is algebraic, namely
It is, for the most part, rested from particularities of semantic contexts and experiential content. And the kind of geometry that we are dealing with is, sorry, the kind of algebra we are also dealing with is ultimately geometric. Namely, it has topological continuous information. And so we see that the majority of the advances that have been made in mathematics since 1960s, 1950s, are within the field or in the continuity of algebraic geometry.
So, let's start with a little bit of introduction about the origin of algebra. Even talking about the origin of algebra is, I would say, is quite a precarious mission in the sense that the origin of algebra is not coming from any specific point or location in the history of mathematics or science.
You can see the first manifestations of algebraic notations in Babylonian manuscripts, Indian manuscripts and Chinese manuscripts. So with that said, with the understanding that the origin of algebra is quite obscure and shouldn't be seen as a specific point in time and history. Nevertheless, I want to kind of bracket this origin to at least a more canonical from a a historian perspective, period. And this is the rights of scholastic Islamic philosophy.
Two thinkers that I want to cover, of course, and again, this is already a bracket within a bracket because there were so many people who contributed to what you might call to be the modern form of algebra. These two people, one is Al-Khar Azmi, who's credited as being the modern father, or at least in kind of a modern grandfather of algebra. And the other one, which I think is more obscure, but nevertheless I think his contributions were even more fundamental
than Al-Khara's, and his name is Tabith Ibn Kura, Tabith Ibn Kura, or Kura. So I'm going to talk about these two Islamic figures, or Kreb said with the advent of modern algebra and show that's how simply by investigating geometric problems they become capable of developing a new system of mathematics or if not developing a new system of mathematics at least put mathematical developments
orienting it toward what you might call to be modern algebra or more accurately they came up with the conceptual basis of algebra so is there any question that you want to ask about some particular aspects of algebra and stuff by the way I will at the end I will talk a little bit about you know the rise of
the modern algebra not modern algebra as it stands today but the rise of modern algebra I'm going to talk about a little bit about Jean Robert Agronde particularly his so-called Agronde-Wesel's diagram which algebra a completely new power as I mentioned algebra is also considered to be the science of magnitudes and magnitudes can be expressed by numbers now in so far as the kind of numbers systems that we have in from antiquity
to the Middle Ages are you know just national numbers rationals so on so forth we do not have something like a complex numbers and agron and Wessel came up with the idea of the so-called complex numbers which are comprised of the so-called imaginary units and that revolutionized the field of algebra completely rested algebraic expressions the science of magnitude from its arithmetic roots and its geometric roots even though a grand famous essay the
invention of imaginary units and complex numbers used the method of geometry and trigonometry so before I start any any question any if you want like more information any observation Jake I can't hear you I turned on the turned off the wrong thing I was just gonna ask like I don't make what makes the square root of negative one uniquely student for like creating another axis of numbers like is it possible to
construct a different kind of complex line, if that makes sense, there's something else that I could equal that is not resolvable. Can you elaborate on this question? I know where you are coming from, but what is exactly the question? I mean, so you are talking about basically imaginary units. The idea that given the fact that, for example, like quadratic equations or cubic equations, they can't deal with negative quantities. Negative quantities or negative magnitudes are basically the sore thumbs in basically
pre-modern mathematics. And there is no such a thing as, you can't see any form of analysis of what negative quantities are until the time of Kant. Kant and Hegel actually started to think about what these things are. And Ogrand is the person who has started to think about negative quantities not as oppositions to the positives, but as their lateralities. Okay. Is it their actualities? Okay, I will talk about this. What are magnitudes? What is the definition of magnitudes? Does
anyone know what magnitude, I mean the most basic philosophical definition of magnitudes. What are magnitudes? Differences in quantity? Quantities are being expressed in terms of magnitudes. are being expressed in terms of magnitudes, not the other way around. Part of me wants to use the word scale, though I can't put it into a definition.
I'm thinking of scale and scalar in terms of magnitudes as opposed to directions, but I haven't got a definition. Magnitudes, actually, it's really quite interesting that Agron had in his famous essay, there's thought experiment, which of course is quite a Kantian idea, that he tries to express magnitude in terms of the scale. By scale, I do not mean a scale like levels, but really a scale, an instrument for balance and counterbalance. Now the thing is that Grossmann apparently has given the most concise, Hermann Grossman, who was a mathematician who is now being regarded as criminally undervalued mathematician of the 18th century, and his contributions are
vastly being reinvestigated. He gives a very brief definition of magnitudes, which has become the most canonical definition mathematics. Simply, two things that can be said to be equal or not equal in one and the same subject. Now, Kant in fact has something also like this in mind when he tries to define negative magnitudes. He makes a famous example of a ship sailing on the ocean being driven forward by two wind currents the idea is
that negative magnitudes are not simply opposite of the positive or positive magnitudes positive quantities like the minus one is not simply opposite of plus 2, they are in fact both positive or more accurately positive quantities, completely real. It's the idea that how they join together in one and the same subject that make one a negative magnitude and the other one a positive magnitude. So a ship that for example being moved forward by a wind current coming from southeast and another wind current
coming from northwest now you can see that the amount of distance that this ship has been traversed since time t1 is being determined by the opposition of these two positive wind currents in one subject namely the ship the The directionality, the product of this opposition is what might you call a negative magnitude or a lateral opposition. So, the significance of the square root of negative one, it's something like that operation
will always produce like an interminable dialect of one's or something along those lines. this is absolutely the case the whole idea is that you know the invention of zero zero is a machine in the sense that it is not nothing it is simply a hinge between magnitudes this is really the algebraic notation of zero and why is that algebra couldn't be invented without zero it's a hinge between between magnitudes. It expresses how magnitudes balance and counterbalance one another in what you might call to be an interminable dialectics between weights, between magnitudes, on different
sides. Right, like zero as a zero. Yes. From an arithmetic perspective, which was dominant from antiquity to end of Middle ages, negative quantities were considered to be simply inferior to positive numbers, just because they come under it. You know, minus one is under plus one, so it's inferior. They weren't even, sometimes they weren't even considered to be real numbers, precisely because of this metaphysical inferiority. But the idea, the systematic understanding of zero in algebra allowed mathematicians to think that it's really not about opposition, simple opposition, it's about how different
positive magnitudes come together, influence one another, this dialectical balance and this balance within one single device zero so our grounds diagram of complex number or so-called I ground Wessel diagram is that so the you know the axis X represents real number and it goes you know from plus whatever to plus whatever and then you have of course the oppositional vector of it the canonical opposition which goes on the minus basically direction both negative
quantities that are being defined in terms of minus sign imaginary number or imaginary units are not really in opposition to the vector x to the axis x is the axis y it's a vertical axis I think that's what always confused me is it is exactly that's the geometric It can show the z-axis and with respect to the real line What's the difference between like a y and a z-axis because you always like having something that? Creates a plane with the number line like at a certain operational point
And I just didn't understand like what the fundamental significance of the square root of negative one was but this like totally answers that Let me I mean this is this is how it goes well for those of you I was going to add one thing real quick. Sure. It's the whole thing with imaginary numbers, which makes it convenient for forming another axis, is the whole fact that it's a square root, and with the complex graph, it multiplying something by i causes it to rotate around. Yes, absolutely. This is what I meant by the power of laterality rather than the power of opposition.
And this is already manifest in Kant's example of how he defines a negative quantity in the sense that the amount or distance traversed by the ship is really a vector that's, you know, and this is the whole idea of vectorial analysis put forward by Grassmann, that it It is no longer simply back and forth opposition like a pendulum, but it denotes a form of orientation or rotation. And a great, great example of this rotation, literally a rotation, is not in Aagrand or Westville, it's in Hamilton's idea of quaternions.
I was thinking that negative can mean and often means a change of direction as in vectors or even on the real line where you're moving left or moving right or rotation when you introduce the square in complex numbers. I was going to say, could you elaborate a little about quaternions? I don't know if everybody, non-mathematicians, understand what they are. Okay. Actually, quaternions is a really difficult concept.
I think the best thing is that I will send you a link that there are some really good articles about quaternions and trying to make them intuitive but for now imagine I will give you a mechanical example of what a quaternion is does any of you know anything about a gimbal it's a device being used in a gyroscope that allows you to so you have XYZ coordinates now Now you can, you know, which of course yields a Cartesian coordinates.
Now quaternion is a form of number on this, you know, the product of XYZ axis that allow you to see the so-called plotting or coordination, you know, it's a form of number that you can't simply achieve it by connecting XYZ of the axis and yield a specific number. It's a form of number in which every time that you are moving from one number to another in the process of succession, you have to rotate XYZ exactly like the kind of a 3D
Vector that you get in your software and how you have to plot and you have to you know rotate the gimbal It's exactly like that. It has a specific pattern which is a rotating a spiraling pattern So it goes like this this is the you know This is one, this is minus one. Let me share this screen.
Thank you. so this is a complex number a is a real number B is called the imaginary component and I which of course is a real number again and I is called imaginary units and imaginary units is that
you can think of this minus one that even under power radicals you get minus one and this is basically the so-called I've run vessel plane or surface or graph whatever you might call it shows what a complex number is and the idea is that for example if you have something like this kinds of equation X 2 plus 1 equals to 0 no you can't really solve this equation basically you can't find an answer for this in natural
numbers or real numbers or any other numbers other than complex numbers which they use basically this imaginary units questions one thing that I thought of is it's just Like how in the past negative numbers were seen as inferior numbers. I mean it would be important to not think of complex numbers as like inferior, you know?
Oh no, no, no, they are not by any means. And the whole idea of imaginary is just an unfortunate choice of words. Imaginary doesn't mean as opposed to real. is actually even more real than real numbers from an ontological perspective yeah I mean the whole progress you know it's a kind of a Russian nested doll of number systems that we have it's
natural integers the rationals the reals and the complex and the idea is that from this point forward we have analytical density between numbers namely within natural numbers between 1 and 2 you can't find any other number but from you know QZ RC integers rationals reals and complex the idea of analytical density it means
that you can find infinite amounts of numbers between any two numbers. Does that apply to integers though? Sorry? If it's between any two numbers, what about integers? Oh sorry, no no you are completely right. I, this is, one second, let me, this should be, this is the box. Nine Queen Zulu ruled over, I forgot the C.
This is basically if you want to memorize the sequence. Isn't the sequence, shouldn't it be NZQRC? No, I think it's NQZ. 9 Zuluquins or let me see so we have nationals we have integers we have Zalem and then rationals are Q. Rationals are Q yes so it should be 9 Zuluquins yes
Basically complex numbers as you see what they differ from every other number is really because of the so-called imaginary units. Basically we are dealing with a new axis between the negatives and positives. And so based on that if we were just trying to like compactly say what the complex numbers add relative to the reels, it's that we can find with the reels like planes or three spaces or whatever, we can describe rotations using a set of numbers because we have like an
imaginary axis makes it a four space and part of that space is not like actual spatial rotation in the three space under discussion but like things that are out of phase whatever that are rotations and we can describe the rotational properties in the real Yes, yeah, although I wouldn't go so much as saying that it's a real three-year space, yes, but yeah, absolutely, yeah. And then, as I mentioned, I will send, I will put this link to Hamilton's Poitonians, so you can see how it works.
And there is in fact this whole idea, I mean there is, you know, there is gimbal in any form of orientational mechanical device, you know, from airplanes to guided missiles and stuff. And the thing is that this is really the kind of algebraic structure that allow, for example, projectile to change its orientation along the geodesic curvature the quaternions are basically this idea of a rotation that they can capture cartesian they're not simply reduced cartes in coordinate but they have
something else they have this rotational axis that is expressed by an imaginary number so should we have a very very brief break before I start far as me and in the Quran for five minutes yeah sure excellent take those names into the sidebar sorry tape those names in the sidebar
and the other one is I think these are, I'm not sure about it because these are Arabic names I think these are I'm not sure about because these are you know Arabic names some but why not be the canonical latinized or
Thank you. Oops, sorry, that wasn't the one I wanted to. No problem. There is, I mean, the first application of quaternions, interestingly enough, was used
in invention of modern telescopes. Because you see, the idea is that, you know, imagine that you have a telescope and you have electrical current that it's you know you have employed a switch for turning off and on and this on and off corresponds to quantities in zero and one now you need to have a negative quantity and this negative quantity basically responsible to not just two coordinates moving left and right of your telescope but also be capable of rotating it according to left and right zooming in and zooming out of the telescope quaternions were used in a way of solve
this problem basically moving the telescope not only along the canonical cartesian coordinates but also giving its rotation according to its coordinates hmm Should we start?
Yeah, let's go ahead and continue. Okay, so very briefly, you know, Farazmi's main contribution to algebra or to algebra is this famous book Al-Jabr and Al-Muqabla which roughly translates to complementation and reduction. Al-Jabr means to add or to complements and means to reduce suggesting two methods of complementation
and reduction for solving algebraic notations algebraic expressions and equations by solving I mean that these two what hard as we take to be universal methods can be applied to the most complex forms of algebraic notations at least complexity its own time and they are capable of reducing the complexity of this mathematical algebraic notation to a simpler notation that can be solved using
existing methods of its time. So as I said, the meaning of jab-r and algebra is coming the word jab-r. was just like when if you're a Farsi speaker Jabra is the word for algebra there is no words I mean the algebra and doesn't take into account the idea of reduction so which is another important method of algebraic solutions it's kind
of like so the word algebra is not by any means the totality of the algebraic method it's simply a you know a convenience westernized you know word that has been used by you know the scholastic philosophers Western scholastic philosophers of Middle Ages to refer to do over all methods of Al-Qarazmi Khara's me. The meaning of the word jabr in Khara's mathematical three ties is adding
equal terms to both sides of an equation in order to eliminate negative terms, reducing the complexity of it. less frequent meaning of the word jabber is multiplying both sides of an equation by one and the same number in order to eliminate fractions because we know that you know I'm sure that you some of your as not we were looking at In the early sessions they had fractions. Fractions are nuisances in mathematics, particularly given the tools of the scholastic philosophers
or a mathematician of antiquity. Fractions are considered to be inferior forms, they're pests in mathematics of that time. So you have to come up with a form of simplifying fractions, simplifying proportions, bringing them from a philosophical perspective of middle ages into perfection, the actual number, the one, two, three, so on, so on, so on, so on, so on, so on, so on, so on, so on, so on, so jamber not also means multiplying size an equation by one and the same number in
order to eliminate fractions so for example if you have one over 13 13 multiplied by 13 and then you get one. In Al-Jabr wa'l-Muqabla Khara Azmi's treatise, the word The word muqabili, let me type these. The word muqabili is...
the whole book that was reading about it sorry I pasted the name for the whole book because I was I was looking at it huh I think that there is a good translation of al-jabr al-mukhabale I think by this guy called George Saliba probably is hard to find translation but if you want to look into it you might be might be able to find it on internet archive where they post you know kind of old manuscripts
it's nevertheless um the meaning of the word is an Arabic word that not only means reduction but also opposition so basically for example you know the word mohobele can be used in terms of war that I am implementing mohobele or against your forces. So it is a reduction in terms of opposition.
By means of opposition, namely positing something that can oppose what has already been posited, I can reduce it and simplify it. So, but the canonical, the one that Khara's meme has used in algebra and formulae, it means reduction of positive terms by subtracting equal amounts from both sides of an equation.
there are also within within the same period of scholastic science and philosophy there are other Arabic or generally Muslim mathematician who simply take the term I'm horrible to mean to equate to equate Named in the sense of comparing opposite sides, comparing opposite sides, the actual meaning of the word muhabbala, opposition, not reduction, opposition.
The combination of the two words algebra and muhabbala is sometimes used in a more general sense performing algebraic operations can also just mean the science of algebra now let me give you some examples of use of these words in the work of a far as being there is this paragraph in a far as meaning it's from page 35 of roses translation of our as this book he says I have divided ten in
two two portions I have multiplied the one of the two portions by the other after this I have multiplied one of the two by itself and the product of the multiplication by itself is four times as much as that of one of the portions by the other. Now it sounds like more perplexing than the Oedipian puzzle. So it's simply a literal translation of this algebraic expression.
in his book, Aparazmi calls one of the portions a thing, a shade, a that work with thing and the other 10 minus thing multiplying the two he obtains in the translation of Rosen 10 things minus a square for the square of the unknown thing Shay Khuras me uses the word mall which means something like well or property it finally obtains the equation a square which is equal to 40 things minus four squares a square look at the algebraic notation a square
which is equal to 40 things minus four squares so the word thing in a haraz means manuscript means wealth or basically what you have mall in Arabic and farce means what do you have? Wealth, think, shade. Notice that we are not using the word unknown, that x does not represent the word unknown here. A square which is equal, so he says that, the one that I mentioned, paragraph that I mentioned it reduces is simply this compact form a square which
is equal to 40 things minus 4 squares the algebraic expression of it is x power or two equals to 40 X minus 4x power 2 next he says the up he basically employs the method of al-jab the method of completion adding the 4x2 to both sides therefore obtaining so if you add to both sides 4x2 this is the method of
al-jab the method of completion you get 5x2 so 4x2 plus 1x2 5x2 and this was minus so it neutralizes this so that on the other side of equation you have 4x 40x which then translates into takes to 8x 40 divided by 5 which then leads to this x equals to 8 so this is the
It's, which of course again being expressed in terms of wealth or shape or things that you have. Now he uses the method of al-mogabale to solve this equation. As I said al-mogabale means to reduce in Chorazmi, but the general meaning of it, it means that
to equate by means of applying opposition. Now using the method of al-Muqabalat, this equation is this. what are we opposing to the both sides of the equation yes so in the introduction to his
turdites Khara's me says that's Imam ma'amun has encouraged me to compose a imam ma'amun is from he basically was a caliph of Arabic Caliph who for the first time it was a kind of famous patron of science and arts but also from a political standpoint it was extremely important because it was the first Arab Caliph who started to become a patron of the Shia minority in
the Muslim world. So he says that he has encouraged me to compose short work on calculating by completion and reduction, confining it to what is easiest and most useful in arithmetic. You see he does not use the word algebra. As I said, at this point we do not have anything like algebra. This is not algebra really in the modern sense, but it's an algebra driven by arithmetic and geometric ambitions, in this case arithmetic ambitions, such as men constantly require in cases of inheritance, legacies, partitions, lawsuits, and trade, and in all their dealings with another over the measuring of lands, the digging of
channels geometrical computation and other objects of various sorts and kinds are concerned now in the first part of our as me and and this is really he already knows the application of the even the most rudimentary form of algebra the arithmetic form of algebra. And as you see, and also as I mentioned at the beginning, and so far as we are working with many unknowns, and if we have many unknowns and only some limited givens, some limited data, are we going to solve these problems?
So you see that algebra before became kind of like a sovereign mathematical field and becoming valorized by virtue of taking mathematics away from experience was important not because of its abstract power but because of its applied power. And this is what, as I read the paragraph by Khawr Azmi, is implying that basically the power of this new method is in its application, where you have too many unknowns and only a limited amount of the givens, limited amount of data. Now, the full title of the Three Ties is, sorry,
I was saying that in the first part of Algebra Al-Muqabili, Khwarazmi explains the solution to six types of algebraic expressions to which all linear and quadratic equations can be reduced. Let me write down these six which of course any advance in print algebra being made throughout middle ages can be simply seen as an extension of our harassment classes of these types of solutions. One is AX
natural coefficients in modern lexicon. Now Khair Azmi gives rules for solving these equations. He presents demonstrations of the rules and he illustrates them by worked examples. In the second chapter of Al-Jabr wa'l-Mu'abalih, Khara's me is concerned with something called a mensuration.
This chapter consists mainly of rules for computing areas and volumes. So you see, I mentioned, if you remember, I used this word, menciration, when I was talking about Urim and his idea of basically areas being the envelopments of the horizontal and the vertical, the manifestation and remission of forms. And despite all the originality of Wurim, that you can see that basically he couldn't arrive at any of those revolutionary insights when he has started to think about, coherently,
about the system of quantification, meanness speed, uniform acceleration, without the contribution people like parents me you know other Arabic Muslim scholastic philosophers for finding for instance the area of a circle is found by multiplying half of the diameter by half of the circumference for finding the circumference three rules are presented if the diameter is D and the periphery P the rules are P and 7 D P equals to 10 to
P six to eight now the rule seven is due to Archimedes who proved that P is less than three one over seven times D and more than three ten over seventy one times d. The same rule seven is also given by another antiquity mathematician, Hieron of Alexandria, in his metricama. In Hebrew, it's mischot ha-midut, edited and translated by Solomon Gans. The rule is also found in chapter seventh of ancient
Indian manuscripts Brahma Gupta basically the first chapter of and I will talk about this but I won't go to this basically did the significance of the second chapter of the first chapter of Al Javra gives a general basically a method of solving algebraic equations the second one he establishes a connection between circles geometry of circles and method of
solving algebraic equations now if you remember the previous sessions the method that we introduced for solving quadratic equations was not using simple it was using rectangles and in fact throughout the Middle Ages particularly as being promoted by Muslim scholars, the hegemonic model of solving, the hegemonic geometric model of using algebraic equations was using rectangles.
So Kharaazmi establishes a new form using circles. And in fact, any method, any algebraic equation that you can use using rectangles by simply using a new square, an additional square, a so-called excess square, or cutting a square off of it, the so-called reduction square, you should be also capable of solving the equation you use rectangles and the method of completing a square using circles and this is really the gist of far as me second chapter which gives
even more accuracy to the solving of algebraic equations I don't go to the details of why gives me more accuracy but you can read about the second chapter of Al-Farras Mead's equation, there are some really good books on this, I will try to upload them. Now, back to, you know, the main scope of Farazmi's Al-Jab, Al-Muqabale, which as I
mentioned it tries to provide general but also more detailed solutions to three types of mixed quadratic equations in for his own words the first type reads roots and squares equal to numbers for instance one square and 10 rooms of the same amount to 39 dirhams the Arabic money currency that is to say what must be the square which when increased by 10 of its own
rooms amounts to 39 the solution is you have you have the number of roots divide roots by two which in the present instance yields five this you multiply it by itself the product is 25 add this to 39 the sum is 64 now take the root of this which is 8 and subtract from it half the number of the roots which is for the remainder is 3 this is the root of the square you thought
are you thought for the square itself is nine now in modern mathematical notation it would be this this as I said one square and ten roots ten roots one square and 10 roots of the same amount to 39 dirhams
39 dirhams that is to say what must be this square which when increased by 10 increased by 10 of its own roots in the X amounts to 39 so he says that the solution is that you should have the number of roots which in the present incense yields five
this you multiply by itself the product is 25 so the next step according to this instruction is x plus 5 to 39 plus 25 equals to 64 the sum at this to 39 the sum is 64 now take the root of this which is 8 and subtract from it half the number of the roots which is for the remainder is 3 so it goes like this
a B the side of which is the desired root X on the four sides of it he construct the rectangles each having one fourth of ten one fourth of ten or two one of one or two as their breath now the square is something becomes
something like this Now the square together with the four rectangles is equal to 39. the ones that you know we just we're talking about 39 now in order to come player complete this where the H the new square that we constructed call it the
we must add four times the square of 2 1 over 2 the ones that we added to each side that is 25 So the area of the largest square is 64, which we showed by algebraic notation, and its sides are 8. So if the area is 64 of this new square, each side is 8.
hence the original the side of the original square the square a B is 8 minus 5 equals to 3 so So as you see, within the scope of what I gave in terms of Khara's
miscontribution, nothing is revolutionary new. All of these methods have already been introduced in antiquity by the Greeks, by the Babylonians, by Indians, the method of completing a square. The genius of Khwarezmi is that he's capable of synthesizing them and coming up with a universal method of dealing with these kinds of mathematical problems, algebraic problems.
for those of you are a little bit confused by these rather simple but no less eccentric treatments of algebraic notation I will get back to this again kind of like a flashback to the Babylonian time and the start to build up again towards me and show how these are all connected I just want to first go over like a broad scope of these two most important work in pro to algebra and then come back and talk a little bit more in details now
Tabithibne Korah is a rather obscure figure even in Muslim world precisely because he was excommunicated, he was basically deemed as a heresyarch. So they managed to get rid of his manuscript, burn them, and... But fortunately before his death and his communication and burning of his books, his main manuscript and his writings on portal algebra reached Western civilization. They were instantly taken off by scholastic philosophers of Middle Ages.
He's also famous, of course there is no trace of this anymore, just the small portions of it's that after Procluse it is famous that he gives the most incisive commentary on Euclid's geometry so we are dealing with a person who knows Euclidean geometry through and through and he's going to employ the Euclidean method in the most elegant way to solve quadratic equations and from at least from the perspective of deployment of
geometric method is far above far as means method which I as I mentioned not is nothing there is in from a methodological perspective there is nothing revolutionary about far as me it's a matter of how he integrates them because the method of completing a square the way that we saw it is very very similar to the Indian method that was dates back to 600 BC so as I mentioned precisely because his books were burned the only version of his
three times and proto algebra with relation to Euclid's geometry is surviving in Latin form it's called the Domuto Octave Esper the astronomical works also translated as the astronomical works of Tabitha ibn al-Khura where he gives a cosmological model of eight spheres of fixed stars. Inside his sphere,
one has to imagine seven spheres of the moon and the sun and the five star planets. Now, In modern astronomy, the fixed stars are soon to be nearly at rest at the equinox to have a small retrograde motion with respect to the fixed stars. The precession of equinoxes, as we call them. Now in Ptolemaic system, the equinoxes are fixed and the stars are supposed to have a slow forward motion of one degree in 100 years now tabit tabit theura notice that this is small amount is not confirmed by the observations the motion of the stars with respect to the
ecognoxious has to be much larger at least in the time after ptolemy if one accepts the very accurate observations made under the reign of Al-Ma'mun, the same that Kelet was mentioning. To explain this, Tabith assumed an oscillatory or periodic motion of the sphere of the fixed stars, the so-called trepidation. Another phenomenon which Tabith wanted to address in this manuscript was an alleged decrease of the obliquity of the ecliptic. The ancient Greeks had used a rough estimate of 24 degrees.
Ptolemy had used a slightly smaller estimate due to Eratosthenes, and the observers at Baghdad's Astronomical Foundation had found that still a smaller obliquity, namely 23 by 33. Now, so within this kind of like that he had noticed observational incompatibilities within the astronomical system of his time in the sense of the Taurmaic system. In order to show that where these incompatibilities, observational incompatibilities with relation to the mathematical model coming from,
he started to develop his own mathematical model and this mathematical model is what you might call to be a true proto algebraic system. He wasn't a mathematician like Al-Kharazmi from this perspective, he was an astronomer. An astronomer who sees an incompatibility between observation and the unemployment of the mathematical model, looks into the mathematical model, sees its insufficiency and then goes on to develop a new mathematical system.
So this new mathematical solution that Tabith develops is also called geometrical verification of the solution of quadratic equations. or the chapter chapters name is under that a specific chapter where he addresses the development of this new mathematical system to address the problem astronomical problems under verification of problems of algebra by
geometrical proofs so there's a paragraph from this particular chapter you remember the word wealth or shade or mall which was the unknown in one mathematical notation so he says there are three fundamental forms or soon roots or elements to which most problems of algebra can be reduced the first basic form is wealth or
model and roots are equal to numbers the way and method of solution by the sixth proposition of Euclid's second book as I shall describe we make a figure the wealth equal to the a square a b g d let me try this for you the wealth
equals to the square a B G D a B G D we make BH equal to the same multiple of the unit in which lines are measured as is in the given number of roots and we complete the area of the edge so first we construct this this is you need to look
at the sixth proposition of the second book of Euclid. Yes the pronunciation is like this. thing well so and also if you want to look at all of the stuff that I was mentioning in terms of you played and that homeworks your first site that you want to look at is proof wiki it has the best resource on the proof of all of the
Euclid propositions with commentaries and stuff quite nicely illustrated so the first basic form is well mall and roots are equal to numbers the way and method of solution by the sixth proposition of Euclid second book is as I shall describe we make this figure the wealth equal to the square a BGT we make BH equal to the same multiple of the units in which lines are measured as is in the given number of roots and we complete the area the H
complete the area the H since the wealth is a BGD the root is clearly a B remember we are talking about a square and its roots this is the area of the square and this is its roots
the route is clearly a B and in the domain of calculation and number it is equal to the product of a B and the units a B and the units in which the lines are measured now a number of these units equal to the given number of roots is in BH hence the product of AB and BH is equal to the roots in the domain of calculation and the number but the
product of a B and B H is the area d H sorry actually let me you let me turn this to G and then we turn this to D so it makes sense so it says that now a number of these units equal to the given number of roots in BH hence the product of AB and BH is equal to the roots in the domain of calculation and number but the product of a B and B H is the area dh
is the area dh because ab is equal to bd ab is equal AB is equal to BD. AB, because we are talking about the square here. You need to look at the proposition 6 in the second book of Euclid. Hence the area of the H is in this way equal to
roots of the problem hence the whole it are gh is equal to the wealth together with the roots now this explanation is quite gnomic from the perspective of modern algebraic expression and the reason for that because it can't equate an area or line segment with a number so he has to go to through the geometric method remember that right now we do not have any way of how to equate an area
with a number we are not we are basically pre Descartes time it's Descartes who starts to associate geometry with numbers we are still in the domain of proto-algebraic geometrical solutions where there is where the connection between geometry and numbers is far from obvious now he therefore introduces a unit of length which we can denote by e if the given equation that he's talking
about in this paragraph is this 2 plus mx equals to n in which x is an unknown number while m and n are given numbers the data he translated into a geometrical equations of this form remember he in a geometrical configuration he introduced a unit of
length which we have here denoted by the letter e now in this second geometrical equation X and e are line segments he then continues by saying that now the wealth and the roots together are equal to no one number so the area gh the whole of your rectangle is known and it is equal to the product of a H and a beam because a B is equal to AG a B is equal to
because this is a square. AB GD is an X square. So AG equals to AB. Hence, Tavid says that the wealth and the roots together are equal to a known number. So the area GH is known. And it is equal to the product of AH and AB. because ab is equal to h so the product of ha and ab is nowhere and the line bh the one that we constructed is nowhere because its number of units is nowhere to it a line ab is added and the product
of HA and AB is known. Now in proposition 6 of book 2 of the elements it is proved that if the line BH if the line BH will be constructed is halved at the point W is halved at the into W the product of HA and AB together with the square of the BW is equal to the square of AW but the product of HA and AB is known and the square of BW is known so the square of AW is known therefore AW is known and if the known
and BW is subtracted a B results as no one and this is the root and if we multiply by itself the square ABGH that is the wealth is no end which is what we wanted to demonstrate I will go through this method that's Tami's uses using the second proposition of the second book next session but you see the way that he approaches this he's still using the idea of a square and your
square is a portion of your wealth in in their lexicon it's basically you're unknown now you either add to it or you subtract something from it here we add to it by constructing the line BH and that allowed us using the second proposition to basically solve the algebraic equation simply using the geometrical method again I will get back to this idea of the two ways the two
more hegemonic ways of solving algebraic expressions or algebraic equations in Middle Ages using the method of adding a square or the method of subtracting a square from your wealth, your unknowns. But of course, as I mentioned, Har Azmi adds a different method, far more complicated, to these two ways of adding or subtracting squares. And that's using circles, particularly drawing concentric circles which translates again to the idea of the square but it's a it's a
more sophisticated and more general solution geometric solution so before I finish today's session and we come back to Tabeet and a little bit more on this whole idea of the algebraic proto-algebra in from antiquity to Middle Ages is there questions or observations I thought it's an interesting question when we were first you're discussing the origins of algebra I was sort of wondering if it was
something that's specifically linked to the development of the alphabet or to alphabetic cultures because that sort of that patient of like a phoneme a graphical shape and a number for arithmetic and certain rules for putting together that are borrowed from spoken syntax or operation to what goes into early algebra but like apparently I was looking in Chinese mathematics which you never used in alphabet was very sophisticated like fairly early on and I just wondering you know being between the development of algebra or or graphical rather contexts that you know might be explained by whether it
comes from an alphabet or from a pictorial well there is this famous essay I have forgotten actually I think I know I mentioned there is this historian of science who wrote extensively that was basically thesis and later he came very famous in the mid 20th century it was this Romanian guy called Sabatai Unguru Sabatai Unguru U N G R U R U and he wrote this essay about you know this
clash between the geometric algebra and the modern notation of algebra symbolic algebra and he started discussing precisely from the perspective that you mentioned that which I mentioned at the beginning of the session that precisely because the algebra the whole conceptual idea of algebra is not graphical this graphically is not simply at the level of metal methodology that's you know algebra today's symbolic algebra is methodologically advanced but it also this change in methodology from arithmetic and geometric to symbolic
the syntactic de-semanified version is actually shift in ontology of mathematics precisely because it unburdens mathematics from experience. Interesting, like, I don't know if you've seen the idea of the history of ontology reweaving space and time together symbolically, something to see as well. yeah a shotley has also something close to this but not as pro formal stance that uh on group
shotley thinks that it's not that geometry loses its traction in transition from proto algebra to modern algebra but it is simply that once symbols are introduced and symbols are just magnitudes you know they're just expressions of the mathematical relations between magnitudes that can no longer be grasped by simple geometric methods he says that once these uh basically these symbols are introduced namely the relations the abstract relations between magnitudes then you see that you are also a stepping beyond the naive intuitions of a space that you have in
Euclidean geometry so rather than algebra completely divorces itself from geometry after the introduction of algebra allows for a different intuition of a space far more advanced and complex than the one presented in Euclidean framework. This is only... but like Aurem, does Aurem have algebra post the introduction of algebra, proto-algebra to the West? It is post. Post, okay. Sorry, Christian, go ahead.
So, this is only tangentially related, but it's very interesting. There's this speculative idea has to do with, I guess if you look at alphabets, the alphabets with vowels are always read left to right, whereas the alphabets without vowels are read right to left. I guess the idea is that right to left, like I forget like which brain is which, which side of your brain, but like it basically has to do with like the efficiency of how we process information. And because languages without vowels, they're more context based. And they're not as linear because you need to intuit sort of the context
in which the word is found to see what the meaning is. Whereas with linear alphabetic scripts that have vowels, there's a possibility for more intense linearization and like sort of fixed abstract, symbolic abstractions. Yes, I mean, I have heard these kinds of arguments, but as you say, they are speculative. I think this kind of argument was like particularly very, very popular, that basically this left to right or right to left, the clash of these is a clash of what you might call to be a clash
of two different systems, two scaffoldings for different potential forms of abstraction was put forward by Derek Kirchhoff. He was a Canadian, I think he was still alive, neuroscientist scientist and cybernetician. Theodor, which book are you talking about? Are you talking about Schatler's book? Was it the Unguru book that you were talking about earlier? Oh, it's an essay. It's an essay. I have forgotten the name of that essay but I think it's got something on
geometrical algebra I will find it for you I will find for you thank you his name is uh this uh i think he's romanian he was romanian and so on gurus thing is that it's really the power of modern algebra is
it's in its operational symbolism and this operational symbolism allows you to identify manipulate and build on the relations between constituting the structures beyond the limits of experience beyond the limits of imagination intuition afforded by geometrical analysis this is just like a really big broad question broader strokes question but how and it might not even be intelligible how how do these ontological understandings shift if not
the ontological understandings I think it's not that the shift in ontological understanding you might call it to be a symptom of shift in the paradigm of mathematical structures and operations that you discover and this is exactly the case in the transition from for two algebra to modern algebra that once you start to look at operations and you will start to deal with the operations in their own terms in their own terms by that in their own terms I mean in pure
abstraction simply using syntactical symbols then you are capable of seeing that there are a difference what you might call level of mathematical structures that for example your previously positive ontological realm couldn't capture but now nevertheless you have positive and then by virtue of that you have shown that there is in fact a more fundamental ontological realm where deeper ontological realm than that your previous assumed ontological dimension. So is it through the resolving of previous contradictions in your... Sometimes previous contradictions and sometimes simply as we see here, sometimes simply by
mathematics more and more from a particular hegemonic current that you might call is assumed to be driving the realm of mathematics or the landscape of mathematics forward and this is really I mean when you look at it the actual the first time that really you have something like algebra is not really this classic philosophy yes far as me Tabitha or are or aim they all have you
know algebraic ambitions in the way that they are approaching mathematics but but simply at the level of conceptual. The first time that you actually come up, you see that someone abstracts this idea of, for example, the relation between geometry and numbers and reinvented on a different level, simply by a very smart form of what you might of what you might call simplification is Descartes this very idea that a line can be represented by two axis bits of X and bits of Y so any thing on this graph on
this plot can be represented by a series of numbers that's a technological fundamental technological breakthrough that allows you to further abstract from the diagram from the geometrical intuitive components simply because you have made explicit a new relationship between geometry the diagram and the number and this is really the course revolution and without the course of revolution you don't get anything of you know of later revolutions in algebra or
calculus simply identify or single out make explicit a new relation between geometry and numbers and we know that this relation between geometry and numbers was already the current the hegemonic current of the scholastic and antiquity the arithmetic and geometric foundations of algebra well once Descartes makes it explicit he also basically this explicitation becomes a
a pivot for a new form of abstraction. Although this idea that, you know, as I mentioned, that these are just some of the more, you kind of prominent characteristics of algebra and how proto-algebra turned modern algebra this idea of moving toward numbers and moving toward symbolic relations after symbolic relations between constituent mathematical
structures there are so many other reasons that you know led to the advent of modern algebra, which I haven't really seen that much historical context for them, I mean, historical analysis of these things, of these causes. Thanks for that. So just so everyone knows, we're at 343 now. So if people want to do last waves of question, otherwise we can end.
Okay, I'll bite. Is this the idea about the word coefficient and the causa efficiens, you know, out of the Aristotle's causal fourfold? Is there anyone who writes explicitly about, like, the elementary parts of algebra, like variables, constants, operations, and so forth, in terms of the four pauses? Like operations as cause of formalis, variables as cause of materialis, that sort of thing? No, I haven't seen, no.
I don't know if it exists. I mean, this is really interesting. But no, I haven't seen anything on this front, no. Cool. Maybe I got a final project idea. Thanks. Also, again, to continue that response to Theodor, the thing is that this whole idea that algebra has a deeper ontological status that, for example, geometry needs again to be taken with a grain of salt and only be seen from the perspective of the philosophical historical reflection. The whole idea of algebra and the revolution of algebra was that it was for the first time
that the mathematical field becomes in previous, indifferent to the discussion about ontological statistics of the mathematical structure. Because you should remember from a historical perspective, it's geometry that for the longest time has these fundamental claims about its ontological priority precisely because it's rooted in intuitions and algebra using symbols becomes imprevious that it's no longer important really you can you can take away all of these attachments this bondage of mathematics to intuitions and still you can have good mathematics for them it's
just basically at this point you need people you know people usually talk about that yeah of course for the hundred years algebra was also a hegemonic to the point of you know ossification of field of mathematics but you need to understand that for hundreds of years it was geometry that was a German and it wasn't tyrannical as you have geometry so all the mathematicians wanted to do wasn't to talk about ontological precedents but simply coming up with a way that they can do mathematics all of this stuff that they could do with the geometric method but no longer being committed to the philosophical ontological
claims of the geometrical analysis so can I ask a question that I'm afraid is like idiotic I isn't isn't it though that mathematics is proven through a empirical usage I think demonstrated is a better word them on empirically demonstrated and empirically demonstrated what what whether what are basically empirical usage demonstrate the efficacy of mathematical structure in encoding the structure of physical reality it does not really demonstrate
anything about mathematics hence the idea of application shouldn't be conflated with the idea of mathematical proof or demonstration that you can in In fact, I mean, this is the whole idea that I do believe that when we are talking about the relation between mathematics and physics, and also the idea of reality, what counts as reality, what counts as being, as existence, we need to, I think, accommodate a local or global deflationary attitude. fields what they mean by real what they mean by existence what they mean as being are need to be understood in terms of local deflationary criteria of their
own fields mathematical entities their reality can be proven but the being of these entities cannot be demonstrated or verified by physics precisely because the being of physics is different from the being of mathematics. Come bunching up these forms of being together we are already creating a bad formula for a naive realism I think. So empirical demonstration, I think, needs to be constrained to the efficacy that all
it testifies to, all it corroborates, is the efficacy of the mathematical structure in encoding the physical structure. It does not say anything, and it shouldn't, about the reality of beings or the ontology of mathematical entities as such. Because the criteria of classification of this ontology, when it comes to mathematics, is determined by mathematics as such. defines as being of an entity. Thanks, that addresses parts of my question.
I kind of like that last part of what you just said because it kind of got into the dialogue being you had last time, it's the illusion is seeing the effect, the practical effect of mathematics emerging merely from the material, the practical effect is almost like an import from the transcendental itself. Yes, yeah, I mean if he wants to be taking side with the transcendental dimension, yes absolutely and this is a quite platonic thesis is really Plato's doctrine and how he sees the
ontological status of mathematics within the realm of transcendentals yes absolutely it is always the reality doesn't come the basically the testimony to the reality of something doesn't come from what you take the reality to be, but from the transcendentals, from the ideals that you think are fictional. Jake, shall we adjourn the session?
Sounds good. Thanks, everyone. Thanks, Silo, for fantastic questions. And Jake and Philip, the question, thank you. Thanks, Reza. Awesome. Thanks a lot, Reza. Thank you very much. OK, see you next week. Sounds good. Take care.