00:00 Intro 01:20 Elements Book 1 Prop 1 - To describe and Equilateral Triangle upon a given finite Right Line. 04:25 Elements Book 1 Prop 2 - At a given Point, to put a Right Line equal to a Right Line given. 09:14 Elements Book 1 Prop 3 - Two unequal Right Lines being given, to cut off a Part from the great Equal to the lesser. 11:10 Elements Book 1 Prop 4 - Theorem 18:24 Elements Book 1 Prop 5 - Theorem - The Angles at the Base of an Isosceles Triangle are equal between themselves; and if the equal Sides be produced, the Angles under the base shall be equal between themselves. 21:20 Problems (logic) with Euclid so far 25:38 Conclusion
Are you thinking about Euclid again? So am I. I believe I can codify all the Euclidean constructions with GEOMETOR Explorer - resulting in a functional hierarchy of all the relationships. Interested?
I read somewhere that Euclid was summarising & systematising 200 years of prior Egyptian knowledge, probably accrued in the field of practical architecture. His books survived the fire at the great library in Alexandria, imagine how much similar intellectual wealth was destroyed. Anyway the effort of revisiting & establishing surer principles in the light of new understanding is as worthwhile doing today as it was back then.
Fantastic. You are easily the best mathematician on TH-cam. I doubt if anyone has ever before described the start of Euclid with such clarity and ease of understanding. Pure brilliance! You said: "I know Euclid is considered to a basis for pure mathematics but that's not really the case. Euclid is interested in describing physical reality, the physical reality of what you can draw on a page with a straight edge and a compass. And in some sense it's an idealized aspect of that reality because in real life lines have thickness and points have some discrete measure otherwise you can't see them. So he's abstracting..." , "...the reliance on pictures, the reliance on our visual understanding, it's not something to be criticized in that context." I completely agree with you, but then what should be a basis for 'pure mathematics' and what exactly does 'pure' mean in this sense? Won't any attempt at purity be inherently problematic in nature because without any foundation in physical reality there will always be disputes between what one person believes they can imagine (such as an infinite set) and what another person believes they can imagine? Would it not be better to abandon all attempts at 'purity' and go completely the other way by establishing a foundation for mathematics based on physical reality? At least then any disputes could be analysed in our shared physical reality and agreement might be reached based on what happens in the real world. For example, two drawn arcs of a circle must be granular in nature and so most of the time they will NOT share the exact same 'granular smallest part' at the point of so-called intersection. Therefore, as can be gleaned from this video, Euclid's approach merely gives us a convenient way to make real world measurements that will be good approximations. Finally I have a request: please, please can you make a video about how the Atomists interpreted Zeno's paradoxes? In those times the concept of a completed infinity was a given absurdity. I've tried to make such a video myself, it's called 'The History of Infinity in Ancient Greece (The Disbeliever, Part 4)', but since I'm a nobody and my videos are a bit amateurish, it means that hardly anyone has seen it. You would do a much better job.
Even considering the issues, you’re doing a massive disservice to your intellect if you don’t at least work through Book I. Not just trying to read and understand it, but attempting each proposition on your own. It’s a marvel in deductive reasoning.
Question on prop.2! Hey, professor, good day! My silly question: "if I use a compass to measure a radius which is equal to the given segment B-C, and then make a circle which has A as the center. Choose any point on the circle, say, point D. Connect A to D, for two points may fix one straight line, then I'll say segment A-D is what we want." I understand that Euclid tried to apply proposition 1 into proposition 2. However, in proposition one, when he made the triangle, he also used " two points determine one straight line". Thank you for great video!! :D
It's not allowed to measure a length with the compass and then carry it over to an other point on the paper. The rule is that you have to collapse the compass every time you lift it off the paper, and so erasing the measurement.
The lecture reminded me of fish. After watching it I had a decent meal of Golden Trout ,153 trout cavear(yes I counted them ) and bread.Then I said, goodbye planet earth and thanks for all the fish.What a life.
The question of area is very subtle indeed, and for that we need to first gain good comprehension of what "flat plain" means. The confluent hypergraph proof narrative between the 1st definition and the "conclusion" of five platonic solids casts the projective shadow of the method of 'Whiteboard and chalk"/"Sand and stick" etc. of the long line mathematicians like Wildberger in his lectures over the whole compilation of Elementa. The planar shadow projection is cast back from the conclusion of the solids and their animated rotations. Greek mathematicians did not view geometry as externalized objects, but as partipatory relation in a Cosmology of the Platonic Solids. Amplituhedron-simplex etc. are returning back to home.
Recently I have been researching Euclid's Elements with a view to finally studying all thirteen books of this classic text. I have been convinced by scholars such as Danielle Macbeth that the Elements was never meant to be an axiomatic system of pure or theoretical geometry. As she puts it: "... one reasons in the diagram in Euclidean geometry, actualizing at each stage some potential of the diagram ... The conclusion that one draws on the basis of a Euclidean demonstration is, for just this reason, contained in one’s starting points only potentially. The steps of the demonstration must be taken to actualize that conclusion. A Euclidean demonstration is not, then, diagram-based, its inferential steps licensed by various features of the diagram. It is properly diagrammatic. One reasons in the diagram, in Euclid, that is, through lines, dia grammon, just as the ancient Greeks claimed ..." (Macbeth, Diagrammatic Reasoning in Euclid’s Elements, p 265). Euclid never actually mentions the compass or straightedge in the Elements, and in describing his geometric constructions he uses unusual forms of the Greek imperative ("Let a circle have been drawn ...") rather than the simple imperative ("Draw a circle ...). I believe the reason for these peculiarities of his method is simply his awareness of the important distinction one must make between geometric objects (points, lines, circles, etc) and our pictorial representations of them. The latter, which we construct with tools such as the compass and the straightedge, are material objects, whereas true geometric objects are abstract ideas. You cannot "draw" an idea with a compass.
@@JoelSjogren0 Yes, in the sense that Plato's Republic exists in heaven, along with all the other Platonic ideas. Some god has created them. So Euclid's demiurge has drawn his circles for him. I believe Proclus describes Euclid as a Platonist. "“Then, if that is his chief concern,” he said, “he will not willingly take part in politics.” “Yes, by the dog” said I, “in his own city he certainly will, yet perhaps not in the city of his birth, except in some providential conjuncture.” “I understand,” he said; “you mean the city whose establishment we have described, the city whose home is in the ideal for I think that it can be found nowhere on earth.” “Well,” said I, “perhaps there is a pattern of it laid up in heaven for him who wishes to contemplate it and so beholding to constitute himself its citizen. But it makes no difference whether it exists now or ever will come into being. The politics of this city only will be his and of none other.”--Plato Republic, 592ab"
@@JoelSjogren0 Yes, in the sense that Plato's Republic exists in heaven, along with all the other Platonic ideas. Some god has created them. So Euclid's demiurge has drawn his circles for him. I believe Proclus describes Euclid as a Platonist. "“Then, if that is his chief concern,” he said, “he will not willingly take part in politics.” “Yes, by the dog” said I, “in his own city he certainly will, yet perhaps not in the city of his birth, except in some providential conjuncture.” “I understand,” he said; “you mean the city whose establishment we have described, the city whose home is in the ideal for I think that it can be found nowhere on earth.” “Well,” said I, “perhaps there is a pattern of it laid up in heaven for him who wishes to contemplate it and so beholding to constitute himself its citizen. But it makes no difference whether it exists now or ever will come into being. The politics of this city only will be his and of none other.”--Plato Republic, 592ab"
The terms used by Euclid are the way they are because he is having a universal conversation.His methods would work in any universe even if it was designed by a knucklehead like the "God" that designed ours.
Excellent comment! Thomas Heath said something similar in his book about the history of Ancient Greek mathematics: Geometry is concerned, not with material things, but with mathematical points, lines, triangles, squares, etc., as objects of pure thought. If we use a diagram in geometry, it is only as an illustration; the triangle which we draw is an imperfect representation of the real triangle of which we think. - A History of Greek Mathematics, Sir Thomas Heath, pg. 286-287.
I'm a professional translator from Classical Greek. Let's give this a closer look: Book 1 Def. 8: ἐπίπεδος δὲ γωνία ἐστὶν ἡ ἐν ἐπιπέδῳ δύο γραμμῶν ἁπτομένων ἀλλήλων καὶ μὴ ἐπ᾽ εὐθείας κειμένων πρὸς ἀλλήλας τῶν γραμμῶν κλίσις. "The angle on plane is two lines on plane touching each other..." Here the meet of lines is defined as actual touching (haptomai). Fine so far on the affirmative aspect of the definition. What about the negative exclusion aspect of the definition? "The descent/lying down/sinking (klisis) of the lines cannot rest (keimai) directly on each other." In other words, the lines can touch each other only once. OK, makes sense. To define a unique angle, the touching lines can't be SUPERIMPOSED INTO SUPERPOSITION except at the single touch. The Heath translation is and utterly incomprehensible confusion: "A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line." The definition does NOT exclude the possibility that the lines can be also curved, as long as they touch each other only once. On further note, the explicit touch is very important here, as the "klisis" has also a strong connotation with dropping down from a higher dimension to lower. The plane of touching can thus be interpreted also as an interdimensional quadrance, which can be "cut" into a spread.
13:57 in-clination. Interesting, klin on Serbian language is angled device for cutting into solid objects like wood - wedge :) Connecting math with etymology :) And not only that, but alao giving pratical purpose to abstract Euclid
Geometry is concerned, not with material things, but with mathematical points, lines, triangles, squares, etc., as objects of pure thought. If we use a diagram in geometry, it is only as an illustration; the triangle which we draw is an imperfect representation of the real triangle of which we think. - A History of Greek Mathematics, Sir Thomas Heath, pg. 286-287.
The problem with non-material things is that one person might claim they can conceive of and theorise about a non-material thing such as a line with no breadth, but another person might contest that we can do such a thing. If one person claims that the imaginary line is infinitely divisible then what is this statement, a theory or an axiom? And if another person claims that we can't actually imagine lines with no breadth, we simply imagine that we can, then how do we settle the dispute? If the basis of geometry or any field of mathematics is non-material then it is based upon nothing more than made up fairy tales in which the prevailing view must be decided by a consensus of opinion.
@@KarmaPeny "The problem with non-material things is that one person might claim they can conceive of and theorise about a non-material thing such as a line with no breadth, but another person might contest that we can do such a thing." Not at all. In Greek Mathematics abstract objects such as point, line, triangle, circle etc. must be able to be reified. That is, a concrete representation of the abstract object must be produced. The Ancient Greeks did that using a pencil, straight-edge and compass. This is the essence of the Platonic theory of Ideas/Forms. The Ancient Greeks never theorized about things that lacked a concrete (or physical) representation. Euclid's definition of a line was poor. A better definition would be a line is that which is formed when a point moves from one fixed point to another. The length of a line is the sense of its size (mégethos). The physical picture of a line is imperfect - lines do not have thickness - their purpose is to help beginners learn about the abstract geometric concepts. "If one person claims that the imaginary line is infinitely divisible then what is this statement, a theory or an axiom? And if another person claims that we can't actually imagine lines with no breadth, we simply imagine that we can, then how do we settle the dispute? " Depends what you mean by infinitely divisible. I would say that any line can be bisected into two parts. This would be an can be verified by straight-edge and compass construction (Book I Prop. 10). Again, Euclid's definition of a line was poor. Lines are not imagined but are defined first and then reified. The line we draw is just an imperfect concrete representation of the perfect abstract line which I have defined previously. Greek geometry is not about compass, straight-edge, pictures or diagrams these are tools used to communicate the abstract Ideas of geometry. "If the basis of geometry or any field of mathematics is non-material then it is based upon nothing more than made up fairy tales in which the prevailing view must be decided by a consensus of opinion." You have a misunderstanding of true mathematics. Mathematics when done properly, is about perfect abstract Ideas/Forms such as line, triangle, circle, magnitude, number, ratio, function, set (not infinite set!), list etc. that can be reified. Infinite sets and infinitesimals are not perfect Ideas/Forms as they cannot be reified. That is, you cannot create a concrete (or physical) representation of an infinite set nor infinitesimal. These concepts are figments of the imagination. Mathematics to the Ancient Greeks was inspired by abstractions of physical reality.
@@vincentdiamond1707 I said: "If one person claims that the imaginary line is infinitely divisible then what is this statement, a theory or an axiom? And if another person claims that we can't actually imagine lines with no breadth, we simply imagine that we can, then how do we settle the dispute?" You replied: "Euclid's definition of a line was poor. A better definition would be a line is that which is formed when a point moves from one fixed point to another." ... "I would say that any line can be bisected into two parts. This would be an can be verified by straight-edge and compass construction (Book I Prop. 10). Again, Euclid's definition of a line was poor. Lines are not imagined but are defined first and then reified. The line we draw is just an imperfect concrete representation of the perfect abstract line which I have defined previously." Your reply demonstrates my point. We have a dispute over the meaning of 'infinitely divisible' and we can't come to any agreement over it! I think we need to include the word 'equal' in your statement, giving "any line can be bisected into two equal parts". My question still stands in that would this be a non-self-evident statement that needs to proven (possibly in the real world with a drawn line) or is it a supposedly a self-evident statement about the so-called perfect line that you assume is a valid concept and I contest is not? You and I both know that for a line to be 'infinitely' divisible then it should always be possible to divide a line into two equal parts. If your drawn (or reified) line consists of an odd number of granular parts then it can't be divided into two equal parts. Can I presume that your so-called 'perfect abstract' line can always be divided into two equal parts? In my opinion, the belief in an infinitely divisible abstract object is just as absurd as believing in an infinite set because neither can be shown to exist in physical reality.
@@KarmaPeny "I think we need to include the word 'equal' in your statement, giving "any line can be bisected into two equal parts". My question still stands in that would this be a non-self-evident statement that needs to proven (possibly in the real world with a drawn line) or is it a supposedly a self-evident statement about the so-called perfect line that you assume is a valid concept and I contest is not?" Euclid's Proposition 10 of Book 1 is a proven statement about the perfect line that can be bisected into two equal parts. The perfect line exists as an abstract concept (or form) and the drawn (or physical) line is its imperfect representation. The straight-edge and compass construction supports or proves the assertion that any abstract line can be divided into two equal parts. Greek geometry is an abstraction of physical reality that is supported by physical constructions! "You and I both know that for a line to be 'infinitely' divisible then it should always be possible to divide a line into two equal parts. If your drawn (or reified) line consists of an odd number of granular parts then it can't be divided into two equal parts." I repeat, what does infinitely divisible mean? It is better to say that any abstract line can be divided into two equal parts. Of course the drawn (or reified) line consists of granular parts. But Euclid's statement is about the abstract line that does not consist of granular parts. "Can I presume that your so-called 'perfect abstract' line can always be divided into two equal parts? In my opinion, the belief in an infinitely divisible abstract object is just as absurd as believing in an infinite set because neither can be shown to exist in physical reality. " Yes, any abstract line can be divided into two equal parts as Euclid demonstrates via a physical construction which supports the statement. Euclid never says that a line is infinitely divisible! That is a false assumption that you have made!
@@vincentdiamond1707 You said; "any abstract line can be divided into two equal parts as Euclid demonstrates via a physical construction which supports the statement." Your description makes no sense to me. What exactly is an 'abstract line'? I've no idea what one is and so how can I verify the claim you make (or that Euclid makes) about it being divisible into equal parts? I can't! Your claim is equivalent to me claiming that an invisible dog can run with infinite speed based on the fact that real world dogs can run quite fast. It is an absurd assumption about an absurd object. You said: "I repeat, what does infinitely divisible mean? It is better to say that any abstract line can be divided into two equal parts. Of course the drawn (or reified) line consists of granular parts. But Euclid's statement is about the abstract line that does not consist of granular parts." I don't have any idea what 'infinitely divisible' means because I don't accept that any usage of the concept of infinity has any tangible meaning. I'm simply using the phrase 'infinitely divisible' as a short way of saying 'the object can be sub-divided into equal parts as many times as we like'. As far as I'm aware, nothing in the real world can meet this criteria because even if an object had this property, it would be impossible to prove (without completing an actual infinity of divisions, whatever that means!). You said: "The straight-edge and compass construction supports or proves the assertion that any abstract line can be divided into two equal parts." No it doesn't. It supports the assertion that a line can only be divided into two equal parts if it consists of an even number of indivisible parts. It says nothing about a so-called 'abstract' line. We can't even determine if it is even valid to talk about an abstract line because I certainly don't know what one is. All I know is that you think that you can talk about it. Obviously we could think about if it would be possible to build a bridge larger than any existing bridge because even though it doesn't currently exist, it is a proposition about something that might physically exist. But I don't accept that we can theorise about concepts that are supposedly partly or wholly somehow disconnected from all physical reality. This would be akin to talking about a spiritual or religious belief. In this respect, I am the atheist and you are the believer, and we all know that people with these opposing viewpoints have very little prospect of reaching agreement about what exists beyond the physical world.
If all concepts in mathematics were arranged in an inverted pyramid structure, then we could always understand higher level concepts in terms of lower level concepts. In such a structure we would need to have a very high level of confidence in the concepts at the lowest level. Personally speaking, I don't have any confidence in 'perfect' abstractions because there's no evidence that such things can exist. And so I'd prefer the bottom level concepts to be based upon real-world objects and actions. I don't understand why I'm called a crackpot for having this opinion and why practically everyone else in the world is happy to accept that mathematics should be based on non-physical things which don't have to be arranged in any top-down structure of dependency. They seem happy to have all sorts of nonsensical and even recursive definitions, all arranged in one huge amorphous mess! Even if I pretend to believe in the 'abstract' nature of non-real-world points and lines, then how can infinitely many points of zero length result in a line of non-zero length? It is such a clear absurdity that I can't understand why everyone isn't shouting and complaining about this. The argument that we can always place more points between any two not-equal points doesn't explain how a non-zero length can emerge or be constructed from zero length objects. Instead of describing lines we might claim to have two fundamental objects: a zero extent and non-zero extent. Another approach might be to claim that linear algebra provides an explanation, but then there's a whole different bunch of definitions and abstract concepts to be 'understood' before we can hope to have any clarity on what exactly a line is. Worse still, all of this abstract stuff is supposedly completely detached from the real world. Don't people realise that their brains are just processing devices? One processing device might take a group of 1,000 fairly random dots as input and its processing might identify a pattern using 30 dots that maps to a human face. A different processing device might identify a pattern using 40 dots that maps to an animal face. Would the processing devices be correct to claim that these patterns exist in the real world or that these numbers exist in the real world or would they be correct to say that patterns and numbers are just internal data items produced by internal processing? People seem happy to claim that patterns and numbers are somehow out there in the universe and so mathematics is, to some extent, discovered. But if numbers and patterns are just things that our minds create and-or use to describe physical stuff (or to describe data derived from physical stuff) then it would be wrong to claim that patterns and numbers have their own existence. If we can be so wrong about the existence of numbers and patterns, then we can also be wrong about being able to imagine non-physical things. Again I know I will get loads of abuse for suggesting this, but this would mean that the whole basis of mathematics has been complete nonsense from the time of Euclid's Elements up to and including the present day.
Karma Peny said "Another approach might be to claim that linear algebra provides an explanation, but then there's a whole different bunch of definitions and abstract concepts to be 'understood' before we can hope to have any clarity on what exactly a line is." But isn't this the approach Dr. Wildberger uses? Do you object to his approach for defining lines and circles as algebraic equations?
@@billh17 You said "But isn't this the approach Dr. Wildberger uses? Do you object to his approach for defining lines and circles as algebraic equations?" I know that the Prof. rejects the concept of 'real numbers' and so I presume he doesn't lay claim to any perfect answers, such as pi or the square root of two, which clearly can't exist. I expect his equations can be used in a computational fashion. So given that we don't know the exact sizes or arrangements of parts of objects at the lowest granular level, the equations might be used in fashion where answers are continually refined until an approximated value is produced that will suffice in the applicable real world scenario. In other words, I don't object to the expression 3x=1 in base 10 if it is clearly understood that an unknown like x might equate to a constant or it might not. If the '1' in 3x=1 can be sub-divided into a number of equal smaller parts that has 3 as a factor, then x can be calculated down to a constant value. If it can't then the equation is not valid. So if 3x=1 is supposed to relate to a real world scenario where the '1' (e.g. one cake) cannot be divided into three equal parts then the equation is strictly invalid but it still might be 'useful' as an approximation (especially where we don't know exactly how many smallest parts there are).
@@KarmaPeny I'm not talking about real numbers and approximations in Dr. Wildberger's approach to geometry. My understanding is that his approach takes points to be ordered pairs of rational numbers (a, b) where a and b are rational numbers. Thus, (2, 5) would be a point. He takes a line to be a linear equation like y = 2 * x + 1. The point (2, 5) would lie on the line y = 2 * x + 1 since it satisfies the equation for the line (where x = 2 and y = 5). He takes a circle to be an equation like x^2 + y^2 = 25, which contains the point (3,4) since it satisfies that equation. In your original post, you seem to be rejecting this approach since " there's a whole different bunch of definitions and abstract concepts to be 'understood' before we can hope to have any clarity on what exactly a line is".
@@billh17 My main concern is that geometry should be able to map onto a real world scenario consisting of smallest indivisible parts. With any collection of rational numbers it is always possible to find a common base in which all the rational numbers can be expressed as terminating decimals. So in this respect, I'm fairly happy with the use of rationals. I'm no expert on his approach to geometry but it would give me concern if there was any way that infinity could sneak in. For example, I'd prefer all linear equations to be bounded (with either fixed or variable bounds) otherwise the implication would be that a line can extend infinitely in either direction. You said "He takes a circle to be an equation like x^2 + y^2 = 25, which contains the point (3,4) since it satisfies that equation. In your original post, you seem to be rejecting this approach since " there's a whole different bunch of definitions and abstract concepts to be 'understood' before we can hope to have any clarity on what exactly a line is"." In my world, I like to think that a point is a smallest part (of space) and a straight line is the shortest number of smallest parts that can be 'measured' from one smallest part to another inclusively. Obviously we need more descriptions to convey relative location when we are talking about more than one point. But I believe these descriptions should be able to correspond to real world objects and actions rather than being 'abstract' things with no physical counterpart. It is the continual movement away from physical reality that I was objecting to in my original post.
Re. original post: I'm with ya, brother! Preach on! 🙌😃 In my own personal philosophical vocabulary, I use the word 'foundationism' to refer to (among some other related concepts) the idea that you are talking about here, namely to try (as much as one can) to structure concepts in one's mind/understanding so that the 'strongest', most 'certain' or 'core' or 'basic' or *'foundational'* concepts are used to support the less 'certain'/'core'/etc. concepts, in a semi-hierarchical (not super-strict, allowing exceptions, since none of us are perfect conceptualizers) structure. The concepts which depend on the the least number of other concepts, yet which themselves have the most support from our experiences of the world that we exist within, are considered more 'foundational' than others. If we find that, within our own minds/understandings, there are concepts which seem to be 'at the base' of many other concepts, yet which *_don't_* really have solid experiential support (perhaps things like wishful assumptions about the nature of the world/universe/cosmos, which we don't really know are true from a wealth of direct experience), then these are perhaps/probably not really strong concepts on which we should be basing so many other concepts. They are 'weak foundations'. When we find a 'weak foundation', the idea of 'foundationism' I'm describing here is to try to 'fix' or rethink/reconceptualize our 'foundations' to try to remove dependencies upon weak foundations and use stronger foundational concepts instead. Or, alternatively, to seek out additional experience which can help us establish whether our prior 'intuitive' foundations are actually 'firm foundations' or not. (This latter alternative is tricky since we are all prone to confirmation bias, tending to 'count' experiences which support our already-believed ideas and to 'discount' those that tend to undermine them. This is where 'foundationism' would branch off into tried-and-true methods like the scientific method, principles of reason and logic, etc. But at its 'core' (i.e. on its *own* foundations) foundationsim is the simpler idea of just looking for the foundations of our own conceptualizations in the first place.) Generally, we try to weed out circular dependencies between concepts when possible, so that the resulting conceptual framework (built on strong foundations) ends up being something like a branching tree with a very strong trunk of foundational concepts and with deep roots in experience. In the end, we will always have to 'assume' certain aspects of even our most foundational concepts (such as the assumption that our 'experience' is 'real' in some sense, and that we are not just 'brains in a vat' being fed artificial sensory inputs for some unknown reason (or no reason whatsoever!)), but I've found that these 'prior assumptions' don't really have to be that many, and for instance don't have to be so numerous as to include things like geometric/mathematical 'lines' and 'points'! Also, we should always recognize that even our 'foundations' can be dramatically _wrong,_ and we needn't be afraid of that possibility, since we are always seeking to find better, stronger foundations anyway! We're always learning, and historically speaking, the incidence of even entire historical societies and cultures operating upon fundamentally flawed 'foundations', which eventually are recognized and either patched up or replaced with 'better' (yet still flawed) foundations, is a very high incidence rate indeed! We shouldn't be the least bit surprised if our *current* society and culture also rests upon flawed foundations that could be improved substantially. Nobody's perfect, none of us, and as a result nor is society/culture either.
Exactly! That is the key to comprehending Euclid and the ontology of holistic source of the projective shadows of geometric intuitions of animated higher dimensional geometry.
What is the main difference between a geometry with straight egde and one with a marked straight edge (ruler)? That was never very clear to me. If you allow to take a length with the compass and transpose it somewhere else in the plane (as we do in practice) some of those complicated constructions shown become trivial. If the aim is practical geometry, why insist on those restictions?
The restrictions enable you to create a universal model that is not confined to your resolution.You can give the model to ant man and he will not have to use your giant ruler.
That's the model that is used. You can't set the compass to a length and then move it somewhere else (in the model). That said, most construction programs (like Geogebra) have a tool that lets you do this. It's a shortcut for doing Euclid's construction. (You can't have the referee hold a point on the first down chain while it's moved across the field!)
Yes, maybe you should start as first and basic problem like on 26:30 that drawn Line has width and therefore looks like area. Similar problem with the Point, which drawn has dimensions and looks like circle. So from first day lesson, math in school make false with saying one but drawing other, and its hipocracy that teachers after that expect from pupuls "exact" answers, when their foundations are not exact. Secondly, I think you should add scepticism were Euclid books his personal invention (like you you use in speak), or it breaks copy rights if Euclid just copy-pasted previous knowledge in his books. So, maybe it was not single Euclid, but plural authors, so if we are not sure, its fair to be sceptic about naming him as responsible.
Omar Khayyam had a number of criticisms about Euclid too, and I'd like to see some of that kind of thing brought out (if his criticisms were interesting enough), because Khayyam was definitely not making his critique from a modern point of view: the sociology of mathematical dispute goes back a long time.
The Ancient Greeks understood pure mathematics to be about theorizing about certain perfect Ideas/Forms such as line, triangle, circle, magnitude, number etc., beginning with definitions and axioms (self-evident truths). Applied mathematics to them was about calculation and construction.
The perfect Ideas/Forms of mathematics are abstract mind-independent objects that exist in a world beyond space and time. What we see in the real world are their imperfect manifestations or "shadows" as Plato calls them in his allegory of the cave. .
I cannot agree with Wildberger's honest opinion that there is something fundamentally wrong or lacking in Euclid. It is his not own fault, but due to relying on translations and interpretation traditions of very questionable value. The gradually degenerating loyalty of translations and consequent inability to comprehend Euclid is the result of the long history of treating Elementa as a schoolboy book (to just mechanically memorize without actual intuitive comprehension) instead of immensly deep foundational study for advanced mathematicians well trained in the self-trasformative art of projective mereological participation in the strictly intuitive ontology of continuous geometry - the deeply spiritual art that Plato's Academy focused on. As set theory destroyed mereology, the consequent academic generations were trained to be professionally incapable of comprehending anything in Elementa, especially the 1st and 3rd definitions. I'm very glad and grateful that Wildberger's long foundational journey is finally returning to mereology with the fresh view of Box arithmetic, and taking the rest of us along.
I most strongly disagree with the last slide that Euclid's compilation of the Greek paradigm of PURE geometry is more "applied math". Pure geometry is called pure because of exclusion of neusis method from pure geometry and considering neusis as applied math for some engineering tasks and such. The context of birth of Greek Pure Geometry is Plato's Academy, and the philosophical quest for Truth and Beauty - which remain the highest values of pure mathematics. Truth theory is not easy, and there is no sign in elementa that it would be committed to what 'mathematical platonism' means in Gödel's definition: postulation of abstract timeless world of Platonia, and correspondence theory of truth with arbitrary axiom of eternal Platonia. No, Elementa is Intuitionism. Brouwer didn't invent it, he just gave Greek intuitive spiritual ontology of pure geometry a breath of life in time of great distress. Yes, it should be very obvious that "line without width" cannot actually exist in any pixelated phenomenology of the external senses. A circle drawn is sand is not a circle in the exact intuitive constructive definition by Euclid etc., the ideal construct can actualize only in the intuitive level of internal sensing. Internal sense of geometric intuition and other holographic presence is how we can empirically perceive the presence of the whole in each part. In Academic physics, David Bohm has brought this comprehension into contemporary public discussion. What Wildberger draws on the method of planar representations are NOT lines and circles, as said. They are just applied math communicative representations of the primitive spiritual and ideal ontology of pure geometry. The truth theory of Intuitionism aka Greek pure geometry relies on Coherence theory of truth, of which Elementa is the prime example. I do understand the hardship of delearning the academic sociological conditioning into materialistic reductionism and reductionistic mathematical physics based on post-modern language games of Cantor and Hilbert. Even more so I admire and am grateful for a great teacher and mathematician whose first revolutionary act against the truth nihilistic absurdity was the book titled DIVINE proportions. Greek pure mathematicians wrestled with their own distributed divinity as creative participants in the Cosmic Demiourge process. They didn't do bad, they stayed humble and loving instead of going crazy in a bad way. No doubt the Mysteries of Eleusis also played a big role in the process of intuiting holistic geometry and then translating some of that into coherent mathematical language.
I just want to point out that pure mathematicians do not seriously accept Euclid as gospel as seems to be implied and are aware of the logical weaknesses. That's why Hilbert updated the axiomatic system (though I personally haven't looked through it, since methods based on linear algebra are more convenient). Personally I don't view most of what came before the later half of the 19th century as rigorous mathematics but only stepping stones to it.
@Terinuva, Actually it is more subtle than that. If you inquire into the foundations of geometry to a modern mathematician, very few of them will take Hilbert's approach seriously. It in fact opens up more difficulties than it surmounts. The default position is that ... geometry is more or less based on Euclid, suitably modernized by a dose of linear algebra. But the actual details are invariably over the hill somewhere.
@@WildEggmathematicscourses Tbh I haven't encountered anybody in my education who has referred to Euclid outside the historical context. If one accepts the description of the continuous via the real numbers (which I recognise you do not) and thus the marriage of our intuitive notions of algebra and geometry, then I think classical geometry is perfectly well described by linear algebra, together with the notions of angle and length given by inner products. The problems you bring up in your video can be solved thusly.
Why do you believe that Euclid meant triangles are equal if the areas are equal (does the translation really say that)? Congruence implies area equality, but certainly not the converse! Does Euclid really mention area?
The video says that there is no prior theory of area in Euclid' s definitions. Did you watch the video or you have problems focussing.He never said what you have in your statement. Euclid was concerned with concepts that would hold in any universe.Area is not a chestnut because it can be perceived as multiple points or multiple lines.
@@antizephdaniel7868 I think Terry Hayes is saying that Euclid proved that the two triangles are congruent (not that their areas are equal). The video is misinterpreting what the theorem and proof does.
@@tdchayes Norman meant if the triangles are equal in that way.Ie based on those angles and sides being equal then the area is also equal in today's mathematical terminology. The conclusion that the area being equal is based on the axiom of equal triangles by euclid on the board. You seem confused.Give a counter proof of what Norman said that makes him wrong.Your claim is baseless.
Prof. Wildberger's criticism of Euclid in this video is no different to the criticism of Bertrand Russell, David Joyce, James Playfair, David Hilbert and a few others that I've read. The difference being that they offered remedies to some of the logical difficulties of Euclid. That is why I believe Wildberger's assertion that "any attempts to pin the foundations of pure geometry on Euclid are unlikely to succeed" is suspect if he doesn't try it himself. For example Euclid defined a point as thus: A point is that which has no part. A better definition (which partly goes back to the Pythagoreans) would be: A point is that which has place or location. It has no size, dimension or extent. Euclid defined a line as thus: A line is breadthless length. A better definition would be: A line is that which is formed when a point moves from one fixed point to another. How do we know when two circles of equal radii intersect? When the sum of their radii is less than the line connecting their centres. What does it mean for a line to be greater than another? When there exists a third line such that when added to the second line their sum equals the first. That is, when you can cut off the second line from the first. What is a triangle? A triangle is a closed connection of three straight lines. What does it mean for two triangles to be congruent (or equal)? A triangle is congruent to another if each side of the first is equal to each side of the second, respectively. Whilst I do agree that some of Euclid's definitions and proofs are vague and dubious, in his defence he was a pioneer in the deductive method of mathematics, who maybe lacked some of the logical insights that we have today. Maybe one day somebody might write a new and improved version of the Elements for posterity just like Euclid.
Euclid' s definition of a point makes more sense than your ramblings of rubbish. A rational point in our world is planck's constant h and it has no parts.It has a size and a dimension. What a fool you are correcting an extradimensional genius like Euclid.
@@antizephdaniel7868 Lol. You are the one that is talking rubbish. Euclid's geometry is not about the physical world but an abstraction of it. Geometry is concerned, not with material things, but with mathematical points, lines, triangles, squares, etc., as objects of pure thought. If we use a diagram in geometry, it is only as an illustration; the triangle which we draw is an imperfect representation of the real triangle of which we think. - Sir Thomas Heath, A History of Greek Mathematics, pg. 286-287. A point is that which has position but not magnitude (or size). - John Playfair, Elements of Geometry. A point is that which has position but not dimensions. - John Casey, The First Six Books of the Elements of Euclid. A magnitude is said to be greater than another by a given magnitude, when this given magnitude being taken from it, the remainder is equal to the other magnitude. - Euclid's Data, Definition 9 A magnitude is said to be less than another by a given magnitude, when this given magnitude being added to it, the whole is equal to the other magnitude. - Euclid's Data. Definition 10. Most of the improvements are not my own but have been taken from various books and articles on the Elements, its history and philosophy that I have read over many years. Therefore, I do know what I'm talking about!
@@vincentdiamond1707 There is no need for this verbose nonsense that you write. To test an abstraction , you apply it to reality, hence my statement about planck's constant.What is the purpose of an abstraction with no use?Euclid s abstractions are universal because they will be validated even if the universe were designed different than what we currently have.
@@antizephdaniel7868 Oh the irony. Your responses so far clearly demonstrates your own ignorance about the Elements and its purpose. The Ancient Greeks understood mathematics to be about platonic Ideas/Forms like circle, triangle, magnitude, number etc. These platonic Ideas/Forms are perfect abstractions that can be reified. Reify means to produce an imperfect concrete representation of an abstract concept. Pure mathematics to the Ancient Greeks was a purely philosophical and intellectual endeavor - an abstraction of physical reality that is supported by imperfect physical constructions. Here is a good introduction to Plato's theory of forms: th-cam.com/video/A7xjoHruQfY/w-d-xo.html
@@antizephdaniel7868 You have no idea how (applied) mathematics and science work. We do NOT apply abstraction (= formal mathematics) directly to reality. The whole point about abstractions is the ability to extrapolate beyond the immediately observable. Once the abstraction yields an interesting conclusion we TEST reality to see if it holds up to the abstraction. If it does the abstraction is "useful" - or a "good theory" in science.
I use the Euclidean plane for the first person of observation of the world. Factorial embedding makes infinity no problem. For modern pure mathematics, what are we even talking about? We both know this is data science. Hence the sociology part.
It is funny how it is so clear that we live in a 4 dimensional Euclidean space yet all books say it is 3 dimensional.What a joke.You'll all end up a cartoon in a cartoon graveyard.
Spacetime, not space, is 4 dimension and the geometry isn't euclidean. Flat spacetime is set in the framework of a minkowski space (this is more akin to a hyperbolic than euclidean geometry). A geometry is only Euclidean if it follows the five axioms as laid out in Euclid. In the geometry of spacetime the parallel postulate is not held (this also means as a consequence the pythagorean theorem and euclidean distance formulas do not hold), and is hence a non-Euclidean geometry. Furthermore space (not space-time) in a flat space-time is a 3 dimensional euclidean space as represented clearly in the metric using a (-+++) signature where ds^2=-c^2t^2+dx^2+dy^2+dz^2 -> ds^2 = dEuclidean^2-c^2dt^2. Furthermore 4-D Euclidean geometry would present a signature of ++++ (or equivalently all +1 along the diagonal inside of the metric tensor), is incompatible with all relativistic effects: time dilation, speed limit of c, and etc all require that the spatial and time components be of different sign. If you prefer to discuss this from finitist perspective replace all differentials with deltas and the arguments still hold.
Here we go again round 2 of an unnecessary and irrelevent attack on Euclid's 'Elements' by a modernist who presumably thinks he could have done it better, an attempt at the restortion of a masterpiece of geometry that isn't damaged. [1] Norman you are not the first mathematician to question whether the arcs intersect, a visiting mathematician at Gresham College did also and gave the mean value theorem as a solution. Unnecessary! All we need to appreciate is that if we construct the square of the base i.e. the square of the radius, then the two arcs will terminate in the corners opposite those that they originated in and MUST THEREFORE INTERSECT. Nothing is added to the proof by embellishment. [2] Considering that all they had back then was a stick & sand box or a stylus & clay tile or ink & parchment on which to draw their geometrical figures, Euclid gave his BRILLIANT DEFINITIONS OF THE PIONT & LINE to stop super bright students including them in the resulting proofs. Looks like he failed as they are still debating them today as somehow having a negative influence on the proofs. WHAT RUBBISH! Euclid's points were rehashed by Richard Dedekind in the 1800's as cuts on the number line and with the advent of visual computer proofs the super bright things will be debating what definition is appropriate for the 'pixelated point and line' for decades yet to come. [3] To help you understand Euclid Norman, lets take 4 SQUARE absolutely identical engineering gauge blocks and assemble the 4 to make a bigger square. The dimension of the square will be double that of a single block and the 'lines' seperating them will form a cross and wont affect the dimensions in any way what so ever yet will still be apparent. The line is not a line but an interface of contact. Like wise where the two interfaces intersect in the middle of the square is just Euclids 'point' which will be evident to us but have no physicality. As I have already said his definitions of the line and point are brilliant. 10/10 Euclid. 0/10 Norman. [4] May I suggest Norman that you might be more gainfully employed revisiting the prime factorisation of the composites starting with the composite of two primes. Maybe you will then demonstrate as I have found after a critical anaysis that there are no shortcuts and no one method shows any advantage over another except simplicity, so it's back to good old fashioned repeated division preferably on a computer with a reduced number field. You should also arrive at the conclusion that manual factorisation is futile and therefore a complete waste of anyones time and should only be carried out by a computer. As for all the factoring sieve methods one reads about but cannot understand, perhaps they should hold 'Factoring Olympiads' to publicly demonstrate which one's works best and their limitations. Enjoy!
NJW----you are doing just fine! I noticed in my work life that everybody standing around thought they could do a better job than the guy actually doing it! LOL
@@acudoc1949 Euclid ---- you did absolutely great man and I honour you. I noticed in my work life that everybody standing around thought they could do a better job than the guy actually doing it! I gave the reply to my comment a thumbs' up.
00:00 Intro
01:20 Elements Book 1 Prop 1 - To describe and Equilateral Triangle upon a given finite Right Line.
04:25 Elements Book 1 Prop 2 - At a given Point, to put a Right Line equal to a Right Line given.
09:14 Elements Book 1 Prop 3 - Two unequal Right Lines being given, to cut off a Part from the great Equal to the lesser.
11:10 Elements Book 1 Prop 4 - Theorem
18:24 Elements Book 1 Prop 5 - Theorem - The Angles at the Base of an Isosceles Triangle are equal between themselves; and if the equal Sides be produced, the Angles under the base shall be equal between themselves.
21:20 Problems (logic) with Euclid so far
25:38 Conclusion
Nice! Thanks!
Are you thinking about Euclid again? So am I. I believe I can codify all the Euclidean constructions with GEOMETOR Explorer - resulting in a functional hierarchy of all the relationships. Interested?
I read somewhere that Euclid was summarising & systematising 200 years of prior Egyptian knowledge, probably accrued in the field of practical architecture. His books survived the fire at the great library in Alexandria, imagine how much similar intellectual wealth was destroyed. Anyway the effort of revisiting & establishing surer principles in the light of new understanding is as worthwhile doing today as it was back then.
Saddens me
Love this, never stop making these educating videos!
Fantastic. You are easily the best mathematician on TH-cam. I doubt if anyone has ever before described the start of Euclid with such clarity and ease of understanding. Pure brilliance!
You said: "I know Euclid is considered to a basis for pure mathematics but that's not really the case. Euclid is interested in describing physical reality, the physical reality of what you can draw on a page with a straight edge and a compass. And in some sense it's an idealized aspect of that reality because in real life lines have thickness and points have some discrete measure otherwise you can't see them. So he's abstracting..." , "...the reliance on pictures, the reliance on our visual understanding, it's not something to be criticized in that context."
I completely agree with you, but then what should be a basis for 'pure mathematics' and what exactly does 'pure' mean in this sense? Won't any attempt at purity be inherently problematic in nature because without any foundation in physical reality there will always be disputes between what one person believes they can imagine (such as an infinite set) and what another person believes they can imagine?
Would it not be better to abandon all attempts at 'purity' and go completely the other way by establishing a foundation for mathematics based on physical reality? At least then any disputes could be analysed in our shared physical reality and agreement might be reached based on what happens in the real world.
For example, two drawn arcs of a circle must be granular in nature and so most of the time they will NOT share the exact same 'granular smallest part' at the point of so-called intersection. Therefore, as can be gleaned from this video, Euclid's approach merely gives us a convenient way to make real world measurements that will be good approximations.
Finally I have a request: please, please can you make a video about how the Atomists interpreted Zeno's paradoxes? In those times the concept of a completed infinity was a given absurdity. I've tried to make such a video myself, it's called 'The History of Infinity in Ancient Greece (The Disbeliever, Part 4)', but since I'm a nobody and my videos are a bit amateurish, it means that hardly anyone has seen it. You would do a much better job.
Even considering the issues, you’re doing a massive disservice to your intellect if you don’t at least work through Book I. Not just trying to read and understand it, but attempting each proposition on your own. It’s a marvel in deductive reasoning.
Makes me wanna go into Euclid again...
Question on prop.2! Hey, professor, good day!
My silly question: "if I use a compass to measure a radius which is equal to the given segment B-C, and then make a circle which has A as the center. Choose any point on the circle, say, point D. Connect A to D, for two points may fix one straight line, then I'll say segment A-D is what we want."
I understand that Euclid tried to apply proposition 1 into proposition 2. However, in proposition one, when he made the triangle, he also used " two points determine one straight line". Thank you for great video!! :D
It's not allowed to measure a length with the compass and then carry it over to an other point on the paper. The rule is that you have to collapse the compass every time you lift it off the paper, and so erasing the measurement.
@@phnaargos Thank you! What a strict rule :D Got it!
Thank you... I'm a law student who just gets mesmerised with greek geometry. Cheers!
The lecture reminded me of fish. After watching it I had a decent meal of Golden Trout ,153 trout cavear(yes I counted them ) and bread.Then I said, goodbye planet earth and thanks for all the fish.What a life.
The question of area is very subtle indeed, and for that we need to first gain good comprehension of what "flat plain" means. The confluent hypergraph proof narrative between the 1st definition and the "conclusion" of five platonic solids casts the projective shadow of the method of 'Whiteboard and chalk"/"Sand and stick" etc. of the long line mathematicians like Wildberger in his lectures over the whole compilation of Elementa.
The planar shadow projection is cast back from the conclusion of the solids and their animated rotations. Greek mathematicians did not view geometry as externalized objects, but as partipatory relation in a Cosmology of the Platonic Solids. Amplituhedron-simplex etc. are returning back to home.
Recently I have been researching Euclid's Elements with a view to finally studying all thirteen books of this classic text. I have been convinced by scholars such as Danielle Macbeth that the Elements was never meant to be an axiomatic system of pure or theoretical geometry. As she puts it:
"... one reasons in the diagram in Euclidean geometry, actualizing at each stage some potential of the diagram ... The conclusion that one draws on the basis of a Euclidean demonstration is, for just this reason, contained in one’s starting points only potentially. The steps of the demonstration must be taken to actualize that conclusion. A Euclidean demonstration is not, then, diagram-based, its inferential steps licensed by various features of the diagram. It is properly diagrammatic. One reasons in the diagram, in Euclid, that is, through lines, dia grammon, just as the ancient Greeks claimed ..." (Macbeth, Diagrammatic Reasoning in Euclid’s Elements, p 265).
Euclid never actually mentions the compass or straightedge in the Elements, and in describing his geometric constructions he uses unusual forms of the Greek imperative ("Let a circle have been drawn ...") rather than the simple imperative ("Draw a circle ...). I believe the reason for these peculiarities of his method is simply his awareness of the important distinction one must make between geometric objects (points, lines, circles, etc) and our pictorial representations of them. The latter, which we construct with tools such as the compass and the straightedge, are material objects, whereas true geometric objects are abstract ideas. You cannot "draw" an idea with a compass.
But can you "let an idea have been drawn"?
@@JoelSjogren0 Yes, in the sense that Plato's Republic exists in heaven, along with all the other Platonic ideas. Some god has created them. So Euclid's demiurge has drawn his circles for him. I believe Proclus describes Euclid as a Platonist.
"“Then, if that is his chief concern,” he said, “he will not willingly take part in politics.” “Yes, by the dog” said I, “in his own city he certainly will, yet perhaps not in the city of his birth, except in some providential conjuncture.” “I understand,” he said; “you mean the city whose establishment we have described, the city whose home is in the ideal for I think that it can be found nowhere on earth.” “Well,” said I, “perhaps there is a pattern of it laid up in heaven for him who wishes to contemplate it and so beholding to constitute himself its citizen. But it makes no difference whether it exists now or ever will come into being. The politics of this city only will be his and of none other.”--Plato Republic, 592ab"
@@JoelSjogren0 Yes, in the sense that Plato's Republic exists in heaven, along with all the other Platonic ideas. Some god has created them. So Euclid's demiurge has drawn his circles for him. I believe Proclus describes Euclid as a Platonist.
"“Then, if that is his chief concern,” he said, “he will not willingly take part in politics.” “Yes, by the dog” said I, “in his own city he certainly will, yet perhaps not in the city of his birth, except in some providential conjuncture.” “I understand,” he said; “you mean the city whose establishment we have described, the city whose home is in the ideal for I think that it can be found nowhere on earth.” “Well,” said I, “perhaps there is a pattern of it laid up in heaven for him who wishes to contemplate it and so beholding to constitute himself its citizen. But it makes no difference whether it exists now or ever will come into being. The politics of this city only will be his and of none other.”--Plato Republic, 592ab"
The terms used by Euclid are the way they are because he is having a universal conversation.His methods would work in any universe even if it was designed by a knucklehead like the "God" that designed ours.
Excellent comment!
Thomas Heath said something similar in his book about the history of Ancient Greek mathematics:
Geometry is concerned, not with material things, but with mathematical points, lines, triangles, squares, etc., as objects of pure thought. If we use a diagram in geometry, it is only as an illustration; the triangle which we draw is an imperfect representation of the real triangle of which we think.
- A History of Greek Mathematics, Sir Thomas Heath, pg. 286-287.
I'm a professional translator from Classical Greek. Let's give this a closer look:
Book 1 Def. 8:
ἐπίπεδος δὲ γωνία ἐστὶν ἡ ἐν ἐπιπέδῳ δύο γραμμῶν ἁπτομένων ἀλλήλων καὶ μὴ ἐπ᾽ εὐθείας κειμένων πρὸς ἀλλήλας τῶν γραμμῶν κλίσις.
"The angle on plane is two lines on plane touching each other..."
Here the meet of lines is defined as actual touching (haptomai). Fine so far on the affirmative aspect of the definition. What about the negative exclusion aspect of the definition?
"The descent/lying down/sinking (klisis) of the lines cannot rest (keimai) directly on each other." In other words, the lines can touch each other only once. OK, makes sense. To define a unique angle, the touching lines can't be SUPERIMPOSED INTO SUPERPOSITION except at the single touch.
The Heath translation is and utterly incomprehensible confusion: "A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line."
The definition does NOT exclude the possibility that the lines can be also curved, as long as they touch each other only once.
On further note, the explicit touch is very important here, as the "klisis" has also a strong connotation with dropping down from a higher dimension to lower. The plane of touching can thus be interpreted also as an interdimensional quadrance, which can be "cut" into a spread.
13:57 in-clination. Interesting, klin on Serbian language is angled device for cutting into solid objects like wood - wedge :) Connecting math with etymology :) And not only that, but alao giving pratical purpose to abstract Euclid
Geometry is concerned, not with material things, but with mathematical points, lines, triangles, squares, etc., as objects of pure thought. If we use a diagram in geometry, it is only as an illustration; the triangle which we draw is an imperfect representation of the real triangle of which we think.
- A History of Greek Mathematics, Sir Thomas Heath, pg. 286-287.
The problem with non-material things is that one person might claim they can conceive of and theorise about a non-material thing such as a line with no breadth, but another person might contest that we can do such a thing.
If one person claims that the imaginary line is infinitely divisible then what is this statement, a theory or an axiom? And if another person claims that we can't actually imagine lines with no breadth, we simply imagine that we can, then how do we settle the dispute?
If the basis of geometry or any field of mathematics is non-material then it is based upon nothing more than made up fairy tales in which the prevailing view must be decided by a consensus of opinion.
@@KarmaPeny
"The problem with non-material things is that one person might claim they can conceive of and theorise about a non-material thing such as a line with no breadth, but another person might contest that we can do such a thing."
Not at all. In Greek Mathematics abstract objects such as point, line, triangle, circle etc. must be able to be reified. That is, a concrete representation of the abstract object must be produced. The Ancient Greeks did that using a pencil, straight-edge and compass. This is the essence of the Platonic theory of Ideas/Forms. The Ancient Greeks never theorized about things that lacked a concrete (or physical) representation.
Euclid's definition of a line was poor. A better definition would be a line is that which is formed when a point moves from one fixed point to another. The length of a line is the sense of its size (mégethos). The physical picture of a line is imperfect - lines do not have thickness - their purpose is to help beginners learn about the abstract geometric concepts.
"If one person claims that the imaginary line is infinitely divisible then what is this statement, a theory or an axiom? And if another person claims that we can't actually imagine lines with no breadth, we simply imagine that we can, then how do we settle the dispute? "
Depends what you mean by infinitely divisible. I would say that any line can be bisected into two parts. This would be an can be verified by straight-edge and compass construction (Book I Prop. 10). Again, Euclid's definition of a line was poor. Lines are not imagined but are defined first and then reified. The line we draw is just an imperfect concrete representation of the perfect abstract line which I have defined previously. Greek geometry is not about compass, straight-edge, pictures or diagrams these are tools used to communicate the abstract Ideas of geometry.
"If the basis of geometry or any field of mathematics is non-material then it is based upon nothing more than made up fairy tales in which the prevailing view must be decided by a consensus of opinion."
You have a misunderstanding of true mathematics. Mathematics when done properly, is about perfect abstract Ideas/Forms such as line, triangle, circle, magnitude, number, ratio, function, set (not infinite set!), list etc. that can be reified. Infinite sets and infinitesimals are not perfect Ideas/Forms as they cannot be reified. That is, you cannot create a concrete (or physical) representation of an infinite set nor infinitesimal. These concepts are figments of the imagination. Mathematics to the Ancient Greeks was inspired by abstractions of physical reality.
@@vincentdiamond1707 I said: "If one person claims that the imaginary line is infinitely divisible then what is this statement, a theory or an axiom? And if another person claims that we can't actually imagine lines with no breadth, we simply imagine that we can, then how do we settle the dispute?"
You replied: "Euclid's definition of a line was poor. A better definition would be a line is that which is formed when a point moves from one fixed point to another." ... "I would say that any line can be bisected into two parts. This would be an can be verified by straight-edge and compass construction (Book I Prop. 10). Again, Euclid's definition of a line was poor. Lines are not imagined but are defined first and then reified. The line we draw is just an imperfect concrete representation of the perfect abstract line which I have defined previously."
Your reply demonstrates my point. We have a dispute over the meaning of 'infinitely divisible' and we can't come to any agreement over it!
I think we need to include the word 'equal' in your statement, giving "any line can be bisected into two equal parts". My question still stands in that would this be a non-self-evident statement that needs to proven (possibly in the real world with a drawn line) or is it a supposedly a self-evident statement about the so-called perfect line that you assume is a valid concept and I contest is not?
You and I both know that for a line to be 'infinitely' divisible then it should always be possible to divide a line into two equal parts. If your drawn (or reified) line consists of an odd number of granular parts then it can't be divided into two equal parts.
Can I presume that your so-called 'perfect abstract' line can always be divided into two equal parts? In my opinion, the belief in an infinitely divisible abstract object is just as absurd as believing in an infinite set because neither can be shown to exist in physical reality.
@@KarmaPeny
"I think we need to include the word 'equal' in your statement, giving "any line can be bisected into two equal parts". My question still stands in that would this be a non-self-evident statement that needs to proven (possibly in the real world with a drawn line) or is it a supposedly a self-evident statement about the so-called perfect line that you assume is a valid concept and I contest is not?"
Euclid's Proposition 10 of Book 1 is a proven statement about the perfect line that can be bisected into two equal parts. The perfect line exists as an abstract concept (or form) and the drawn (or physical) line is its imperfect representation. The straight-edge and compass construction supports or proves the assertion that any abstract line can be divided into two equal parts. Greek geometry is an abstraction of physical reality that is supported by physical constructions!
"You and I both know that for a line to be 'infinitely' divisible then it should always be possible to divide a line into two equal parts. If your drawn (or reified) line consists of an odd number of granular parts then it can't be divided into two equal parts."
I repeat, what does infinitely divisible mean? It is better to say that any abstract line can be divided into two equal parts. Of course the drawn (or reified) line consists of granular parts. But Euclid's statement is about the abstract line that does not consist of granular parts.
"Can I presume that your so-called 'perfect abstract' line can always be divided into two equal parts? In my opinion, the belief in an infinitely divisible abstract object is just as absurd as believing in an infinite set because neither can be shown to exist in physical reality.
"
Yes, any abstract line can be divided into two equal parts as Euclid demonstrates via a physical construction which supports the statement. Euclid never says that a line is infinitely divisible! That is a false assumption that you have made!
@@vincentdiamond1707 You said; "any abstract line can be divided into two equal parts as Euclid demonstrates via a physical construction which supports the statement."
Your description makes no sense to me. What exactly is an 'abstract line'? I've no idea what one is and so how can I verify the claim you make (or that Euclid makes) about it being divisible into equal parts? I can't! Your claim is equivalent to me claiming that an invisible dog can run with infinite speed based on the fact that real world dogs can run quite fast. It is an absurd assumption about an absurd object.
You said: "I repeat, what does infinitely divisible mean? It is better to say that any abstract line can be divided into two equal parts. Of course the drawn (or reified) line consists of granular parts. But Euclid's statement is about the abstract line that does not consist of granular parts."
I don't have any idea what 'infinitely divisible' means because I don't accept that any usage of the concept of infinity has any tangible meaning. I'm simply using the phrase 'infinitely divisible' as a short way of saying 'the object can be sub-divided into equal parts as many times as we like'. As far as I'm aware, nothing in the real world can meet this criteria because even if an object had this property, it would be impossible to prove (without completing an actual infinity of divisions, whatever that means!).
You said: "The straight-edge and compass construction supports or proves the assertion that any abstract line can be divided into two equal parts."
No it doesn't. It supports the assertion that a line can only be divided into two equal parts if it consists of an even number of indivisible parts. It says nothing about a so-called 'abstract' line. We can't even determine if it is even valid to talk about an abstract line because I certainly don't know what one is. All I know is that you think that you can talk about it.
Obviously we could think about if it would be possible to build a bridge larger than any existing bridge because even though it doesn't currently exist, it is a proposition about something that might physically exist. But I don't accept that we can theorise about concepts that are supposedly partly or wholly somehow disconnected from all physical reality. This would be akin to talking about a spiritual or religious belief. In this respect, I am the atheist and you are the believer, and we all know that people with these opposing viewpoints have very little prospect of reaching agreement about what exists beyond the physical world.
If all concepts in mathematics were arranged in an inverted pyramid structure, then we could always understand higher level concepts in terms of lower level concepts. In such a structure we would need to have a very high level of confidence in the concepts at the lowest level.
Personally speaking, I don't have any confidence in 'perfect' abstractions because there's no evidence that such things can exist. And so I'd prefer the bottom level concepts to be based upon real-world objects and actions. I don't understand why I'm called a crackpot for having this opinion and why practically everyone else in the world is happy to accept that mathematics should be based on non-physical things which don't have to be arranged in any top-down structure of dependency. They seem happy to have all sorts of nonsensical and even recursive definitions, all arranged in one huge amorphous mess!
Even if I pretend to believe in the 'abstract' nature of non-real-world points and lines, then how can infinitely many points of zero length result in a line of non-zero length? It is such a clear absurdity that I can't understand why everyone isn't shouting and complaining about this. The argument that we can always place more points between any two not-equal points doesn't explain how a non-zero length can emerge or be constructed from zero length objects.
Instead of describing lines we might claim to have two fundamental objects: a zero extent and non-zero extent. Another approach might be to claim that linear algebra provides an explanation, but then there's a whole different bunch of definitions and abstract concepts to be 'understood' before we can hope to have any clarity on what exactly a line is. Worse still, all of this abstract stuff is supposedly completely detached from the real world.
Don't people realise that their brains are just processing devices? One processing device might take a group of 1,000 fairly random dots as input and its processing might identify a pattern using 30 dots that maps to a human face. A different processing device might identify a pattern using 40 dots that maps to an animal face. Would the processing devices be correct to claim that these patterns exist in the real world or that these numbers exist in the real world or would they be correct to say that patterns and numbers are just internal data items produced by internal processing?
People seem happy to claim that patterns and numbers are somehow out there in the universe and so mathematics is, to some extent, discovered. But if numbers and patterns are just things that our minds create and-or use to describe physical stuff (or to describe data derived from physical stuff) then it would be wrong to claim that patterns and numbers have their own existence.
If we can be so wrong about the existence of numbers and patterns, then we can also be wrong about being able to imagine non-physical things. Again I know I will get loads of abuse for suggesting this, but this would mean that the whole basis of mathematics has been complete nonsense from the time of Euclid's Elements up to and including the present day.
Karma Peny said "Another approach might be to claim that linear algebra provides an explanation, but then there's a whole different bunch of definitions and abstract concepts to be 'understood' before we can hope to have any clarity on what exactly a line is." But isn't this the approach Dr. Wildberger uses? Do you object to his approach for defining lines and circles as algebraic equations?
@@billh17 You said "But isn't this the approach Dr. Wildberger uses? Do you object to his approach for defining lines and circles as algebraic equations?"
I know that the Prof. rejects the concept of 'real numbers' and so I presume he doesn't lay claim to any perfect answers, such as pi or the square root of two, which clearly can't exist. I expect his equations can be used in a computational fashion. So given that we don't know the exact sizes or arrangements of parts of objects at the lowest granular level, the equations might be used in fashion where answers are continually refined until an approximated value is produced that will suffice in the applicable real world scenario.
In other words, I don't object to the expression 3x=1 in base 10 if it is clearly understood that an unknown like x might equate to a constant or it might not. If the '1' in 3x=1 can be sub-divided into a number of equal smaller parts that has 3 as a factor, then x can be calculated down to a constant value. If it can't then the equation is not valid. So if 3x=1 is supposed to relate to a real world scenario where the '1' (e.g. one cake) cannot be divided into three equal parts then the equation is strictly invalid but it still might be 'useful' as an approximation (especially where we don't know exactly how many smallest parts there are).
@@KarmaPeny I'm not talking about real numbers and approximations in Dr. Wildberger's approach to geometry. My understanding is that his approach takes points to be ordered pairs of rational numbers (a, b) where a and b are rational numbers. Thus, (2, 5) would be a point. He takes a line to be a linear equation like y = 2 * x + 1. The point (2, 5) would lie on the line y = 2 * x + 1 since it satisfies the equation for the line (where x = 2 and y = 5). He takes a circle to be an equation like x^2 + y^2 = 25, which contains the point (3,4) since it satisfies that equation. In your original post, you seem to be rejecting this approach since " there's a whole different bunch of definitions and abstract concepts to be 'understood' before we can hope to have any clarity on what exactly a line is".
@@billh17 My main concern is that geometry should be able to map onto a real world scenario consisting of smallest indivisible parts. With any collection of rational numbers it is always possible to find a common base in which all the rational numbers can be expressed as terminating decimals. So in this respect, I'm fairly happy with the use of rationals.
I'm no expert on his approach to geometry but it would give me concern if there was any way that infinity could sneak in. For example, I'd prefer all linear equations to be bounded (with either fixed or variable bounds) otherwise the implication would be that a line can extend infinitely in either direction.
You said "He takes a circle to be an equation like x^2 + y^2 = 25, which contains the point (3,4) since it satisfies that equation. In your original post, you seem to be rejecting this approach since " there's a whole different bunch of definitions and abstract concepts to be 'understood' before we can hope to have any clarity on what exactly a line is"."
In my world, I like to think that a point is a smallest part (of space) and a straight line is the shortest number of smallest parts that can be 'measured' from one smallest part to another inclusively. Obviously we need more descriptions to convey relative location when we are talking about more than one point. But I believe these descriptions should be able to correspond to real world objects and actions rather than being 'abstract' things with no physical counterpart. It is the continual movement away from physical reality that I was objecting to in my original post.
Re. original post: I'm with ya, brother! Preach on! 🙌😃
In my own personal philosophical vocabulary, I use the word 'foundationism' to refer to (among some other related concepts) the idea that you are talking about here, namely to try (as much as one can) to structure concepts in one's mind/understanding so that the 'strongest', most 'certain' or 'core' or 'basic' or *'foundational'* concepts are used to support the less 'certain'/'core'/etc. concepts, in a semi-hierarchical (not super-strict, allowing exceptions, since none of us are perfect conceptualizers) structure. The concepts which depend on the the least number of other concepts, yet which themselves have the most support from our experiences of the world that we exist within, are considered more 'foundational' than others.
If we find that, within our own minds/understandings, there are concepts which seem to be 'at the base' of many other concepts, yet which *_don't_* really have solid experiential support (perhaps things like wishful assumptions about the nature of the world/universe/cosmos, which we don't really know are true from a wealth of direct experience), then these are perhaps/probably not really strong concepts on which we should be basing so many other concepts. They are 'weak foundations'.
When we find a 'weak foundation', the idea of 'foundationism' I'm describing here is to try to 'fix' or rethink/reconceptualize our 'foundations' to try to remove dependencies upon weak foundations and use stronger foundational concepts instead. Or, alternatively, to seek out additional experience which can help us establish whether our prior 'intuitive' foundations are actually 'firm foundations' or not. (This latter alternative is tricky since we are all prone to confirmation bias, tending to 'count' experiences which support our already-believed ideas and to 'discount' those that tend to undermine them. This is where 'foundationism' would branch off into tried-and-true methods like the scientific method, principles of reason and logic, etc. But at its 'core' (i.e. on its *own* foundations) foundationsim is the simpler idea of just looking for the foundations of our own conceptualizations in the first place.)
Generally, we try to weed out circular dependencies between concepts when possible, so that the resulting conceptual framework (built on strong foundations) ends up being something like a branching tree with a very strong trunk of foundational concepts and with deep roots in experience.
In the end, we will always have to 'assume' certain aspects of even our most foundational concepts (such as the assumption that our 'experience' is 'real' in some sense, and that we are not just 'brains in a vat' being fed artificial sensory inputs for some unknown reason (or no reason whatsoever!)), but I've found that these 'prior assumptions' don't really have to be that many, and for instance don't have to be so numerous as to include things like geometric/mathematical 'lines' and 'points'!
Also, we should always recognize that even our 'foundations' can be dramatically _wrong,_ and we needn't be afraid of that possibility, since we are always seeking to find better, stronger foundations anyway! We're always learning, and historically speaking, the incidence of even entire historical societies and cultures operating upon fundamentally flawed 'foundations', which eventually are recognized and either patched up or replaced with 'better' (yet still flawed) foundations, is a very high incidence rate indeed! We shouldn't be the least bit surprised if our *current* society and culture also rests upon flawed foundations that could be improved substantially. Nobody's perfect, none of us, and as a result nor is society/culture either.
22:28
What about common notion 5; "The whole is greater than the part."
Exactly! That is the key to comprehending Euclid and the ontology of holistic source of the projective shadows of geometric intuitions of animated higher dimensional geometry.
I love you man
What is the main difference between a geometry with straight egde and one with a marked straight edge (ruler)? That was never very clear to me. If you allow to take a length with the compass and transpose it somewhere else in the plane (as we do in practice) some of those complicated constructions shown become trivial. If the aim is practical geometry, why insist on those restictions?
The restrictions enable you to create a universal model that is not confined to your resolution.You can give the model to ant man and he will not have to use your giant ruler.
Euclid is the original master of geometry.
So the compass collapses as soon as you lift it off the paper?
That's the model that is used. You can't set the compass to a length and then move it somewhere else (in the model). That said, most construction programs (like Geogebra) have a tool that lets you do this. It's a shortcut for doing Euclid's construction. (You can't have the referee hold a point on the first down chain while it's moved across the field!)
How be we bring back the Pons Asinorum as a pre-requisite for running in a US Presidential primacy?
Or for any political office, for that matter! 😄
This is amazing. Id love to take a course in Euclidean math
Sandy Bultena has books 1-7 covered and Clark.edu has the books with some commentary. See Thomas L heath for the good stuff :)
Yes, maybe you should start as first and basic problem like on 26:30 that drawn Line has width and therefore looks like area. Similar problem with the Point, which drawn has dimensions and looks like circle. So from first day lesson, math in school make false with saying one but drawing other, and its hipocracy that teachers after that expect from pupuls "exact" answers, when their foundations are not exact.
Secondly, I think you should add scepticism were Euclid books his personal invention (like you you use in speak), or it breaks copy rights if Euclid just copy-pasted previous knowledge in his books. So, maybe it was not single Euclid, but plural authors, so if we are not sure, its fair to be sceptic about naming him as responsible.
If you can't even spell the word, you're probably not an authority on hypocrisy in education.
@@TheDavidlloydjones So knowing alien English language makes one less hipocritical?
Is this type of geometry mathematics at all, or is it pragmatic tools for representing it?
No it is not mathematics he just wants you to buy fish on a sunday and celebrate Jesus Christ's great catch knucklehead!
I think Euclid's most fantastic book is V. It would be very interesting to have a video about him
@ r a Yes that is a quite interesting Book, I will have to discuss it at some point. Thanks for the suggestion.
thank you
Thx. Again
Omar Khayyam had a number of criticisms about Euclid too, and I'd like to see some of that kind of thing brought out (if his criticisms were interesting enough), because Khayyam was definitely not making his critique from a modern point of view: the sociology of mathematical dispute goes back a long time.
The Ancient Greeks understood pure mathematics to be about theorizing about certain perfect Ideas/Forms such as line, triangle, circle, magnitude, number etc., beginning with definitions and axioms (self-evident truths). Applied mathematics to them was about calculation and construction.
The perfect Ideas/Forms of mathematics are abstract mind-independent objects that exist in a world beyond space and time. What we see in the real world
are their imperfect manifestations or "shadows" as Plato calls them in his allegory of the cave.
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the II prop is right withe a isocele rectangle rectangle with projection instiade of rrotation and taking the squr two ratio a have a sketsh....
Elliptic curve and algebraic geometry, I learned many things from your writings, Merci !
Thank you Professor!!! Merci beaucoup!!!
What is pure mathematics if not an ideal aspect of physical reality?
I cannot agree with Wildberger's honest opinion that there is something fundamentally wrong or lacking in Euclid. It is his not own fault, but due to relying on translations and interpretation traditions of very questionable value. The gradually degenerating loyalty of translations and consequent inability to comprehend Euclid is the result of the long history of treating Elementa as a schoolboy book (to just mechanically memorize without actual intuitive comprehension) instead of immensly deep foundational study for advanced mathematicians well trained in the self-trasformative art of projective mereological participation in the strictly intuitive ontology of continuous geometry - the deeply spiritual art that Plato's Academy focused on.
As set theory destroyed mereology, the consequent academic generations were trained to be professionally incapable of comprehending anything in Elementa, especially the 1st and 3rd definitions. I'm very glad and grateful that Wildberger's long foundational journey is finally returning to mereology with the fresh view of Box arithmetic, and taking the rest of us along.
I most strongly disagree with the last slide that Euclid's compilation of the Greek paradigm of PURE geometry is more "applied math". Pure geometry is called pure because of exclusion of neusis method from pure geometry and considering neusis as applied math for some engineering tasks and such.
The context of birth of Greek Pure Geometry is Plato's Academy, and the philosophical quest for Truth and Beauty - which remain the highest values of pure mathematics. Truth theory is not easy, and there is no sign in elementa that it would be committed to what 'mathematical platonism' means in Gödel's definition: postulation of abstract timeless world of Platonia, and correspondence theory of truth with arbitrary axiom of eternal Platonia.
No, Elementa is Intuitionism. Brouwer didn't invent it, he just gave Greek intuitive spiritual ontology of pure geometry a breath of life in time of great distress.
Yes, it should be very obvious that "line without width" cannot actually exist in any pixelated phenomenology of the external senses. A circle drawn is sand is not a circle in the exact intuitive constructive definition by Euclid etc., the ideal construct can actualize only in the intuitive level of internal sensing. Internal sense of geometric intuition and other holographic presence is how we can empirically perceive the presence of the whole in each part. In Academic physics, David Bohm has brought this comprehension into contemporary public discussion.
What Wildberger draws on the method of planar representations are NOT lines and circles, as said. They are just applied math communicative representations of the primitive spiritual and ideal ontology of pure geometry. The truth theory of Intuitionism aka Greek pure geometry relies on Coherence theory of truth, of which Elementa is the prime example.
I do understand the hardship of delearning the academic sociological conditioning into materialistic reductionism and reductionistic mathematical physics based on post-modern language games of Cantor and Hilbert. Even more so I admire and am grateful for a great teacher and mathematician whose first revolutionary act against the truth nihilistic absurdity was the book titled DIVINE proportions.
Greek pure mathematicians wrestled with their own distributed divinity as creative participants in the Cosmic Demiourge process. They didn't do bad, they stayed humble and loving instead of going crazy in a bad way. No doubt the Mysteries of Eleusis also played a big role in the process of intuiting holistic geometry and then translating some of that into coherent mathematical language.
Glad to join the finitist cabal of TH-cam, with N J Wildberger as Doge.
I just want to point out that pure mathematicians do not seriously accept Euclid as gospel as seems to be implied and are aware of the logical weaknesses. That's why Hilbert updated the axiomatic system (though I personally haven't looked through it, since methods based on linear algebra are more convenient).
Personally I don't view most of what came before the later half of the 19th century as rigorous mathematics but only stepping stones to it.
@Terinuva, Actually it is more subtle than that. If you inquire into the foundations of geometry to a modern mathematician, very few of them will take Hilbert's approach seriously. It in fact opens up more difficulties than it surmounts. The default position is that ... geometry is more or less based on Euclid, suitably modernized by a dose of linear algebra. But the actual details are invariably over the hill somewhere.
@@WildEggmathematicscourses Tbh I haven't encountered anybody in my education who has referred to Euclid outside the historical context.
If one accepts the description of the continuous via the real numbers (which I recognise you do not) and thus the marriage of our intuitive notions of algebra and geometry, then I think classical geometry is perfectly well described by linear algebra, together with the notions of angle and length given by inner products.
The problems you bring up in your video can be solved thusly.
Why do you believe that Euclid meant triangles are equal if the areas are equal (does the translation really say that)? Congruence implies area equality, but certainly not the converse! Does Euclid really mention area?
The video says that there is no prior theory of area in Euclid' s definitions. Did you watch the video or you have problems focussing.He never said what you have in your statement. Euclid was concerned with concepts that would hold in any universe.Area is not a chestnut because it can be perceived as multiple points or multiple lines.
@@antizephdaniel7868 I think Terry Hayes is saying that Euclid proved that the two triangles are congruent (not that their areas are equal). The video is misinterpreting what the theorem and proof does.
@@antizephdaniel7868 15:44
@@tdchayes Norman meant if the triangles are equal in that way.Ie based on those angles and sides being equal then the area is also equal in today's mathematical terminology. The conclusion that the area being equal is based on the axiom of equal triangles by euclid on the board. You seem confused.Give a counter proof of what Norman said that makes him wrong.Your claim is baseless.
Please do Euclid an honor he deserves. And work precisely with the appropriate tools and rigour.
Prof. Wildberger's criticism of Euclid in this video is no different to the criticism of Bertrand Russell, David Joyce, James Playfair, David Hilbert
and a few others that I've read. The difference being that they offered remedies to some of the logical difficulties of Euclid.
That is why I believe Wildberger's assertion that "any attempts to pin the foundations of pure geometry on Euclid are unlikely to succeed" is suspect if he doesn't try it himself.
For example Euclid defined a point as thus:
A point is that which has no part.
A better definition (which partly goes back to the Pythagoreans) would be:
A point is that which has place or location. It has no size, dimension or extent.
Euclid defined a line as thus:
A line is breadthless length.
A better definition would be:
A line is that which is formed when a point moves from one fixed point to another.
How do we know when two circles of equal radii intersect?
When the sum of their radii is less than the line connecting their centres.
What does it mean for a line to be greater than another?
When there exists a third line such that when added to the second line their sum equals the first.
That is, when you can cut off the second line from the first.
What is a triangle?
A triangle is a closed connection of three straight lines.
What does it mean for two triangles to be congruent (or equal)?
A triangle is congruent to another if each side of the first is equal to each side of the second, respectively.
Whilst I do agree that some of Euclid's definitions and proofs are vague and dubious, in his defence he was a pioneer in the
deductive method of mathematics, who maybe lacked some of the logical insights that we have today. Maybe one day somebody might write a new and improved version of the Elements for posterity just like Euclid.
Euclid' s definition of a point makes more sense than your ramblings of rubbish. A rational point in our world is planck's constant h and it has no parts.It has a size and a dimension. What a fool you are correcting an extradimensional genius like Euclid.
@@antizephdaniel7868
Lol. You are the one that is talking rubbish. Euclid's geometry is not about the physical world but an abstraction of it.
Geometry is concerned, not with material things, but with mathematical points, lines, triangles, squares, etc., as objects of pure thought. If we use a diagram in geometry, it is only as an illustration; the triangle which we draw is an imperfect representation of the real triangle of which we think.
- Sir Thomas Heath, A History of Greek Mathematics, pg. 286-287.
A point is that which has position but not magnitude (or size).
- John Playfair, Elements of Geometry.
A point is that which has position but not dimensions.
- John Casey, The First Six Books of the Elements of Euclid.
A magnitude is said to be greater than another by a given magnitude, when this given magnitude being taken from it, the remainder is equal to the other magnitude.
- Euclid's Data, Definition 9
A magnitude is said to be less than another by a given magnitude, when this given magnitude being added to it, the whole is equal to the other magnitude.
- Euclid's Data. Definition 10.
Most of the improvements are not my own but have been taken from various books and articles on the Elements, its history and philosophy that I have read over many years. Therefore, I do know what I'm talking about!
@@vincentdiamond1707 There is no need for this verbose nonsense that you write. To test an abstraction , you apply it to reality, hence my statement about planck's constant.What is the purpose of an abstraction with no use?Euclid s abstractions are universal because they will be validated even if the universe were designed different than what we currently have.
@@antizephdaniel7868
Oh the irony. Your responses so far clearly demonstrates your own ignorance about the Elements and its purpose. The Ancient Greeks understood mathematics to be about platonic Ideas/Forms like circle, triangle, magnitude, number etc. These platonic Ideas/Forms are perfect abstractions that can be reified. Reify means to produce an imperfect concrete representation of an abstract concept. Pure mathematics to the Ancient Greeks was a purely philosophical and intellectual endeavor - an abstraction of physical reality that is supported by imperfect physical constructions.
Here is a good introduction to Plato's theory of forms:
th-cam.com/video/A7xjoHruQfY/w-d-xo.html
@@antizephdaniel7868 You have no idea how (applied) mathematics and science work. We do NOT apply abstraction (= formal mathematics) directly to reality. The whole point about abstractions is the ability to extrapolate beyond the immediately observable.
Once the abstraction yields an interesting conclusion we TEST reality to see if it holds up to the abstraction. If it does the abstraction is "useful" - or a "good theory" in science.
I use the Euclidean plane for the first person of observation of the world. Factorial embedding makes infinity no problem. For modern pure mathematics, what are we even talking about? We both know this is data science. Hence the sociology part.
Hi Norman. Will you please add the subscript? It would help me a lot.
Wildberger seems to regularly misuse the compass as a measuring tool. All his arguments become bogus when he does that.
It is funny how it is so clear that we live in a 4 dimensional Euclidean space yet all books say it is 3 dimensional.What a joke.You'll all end up a cartoon in a cartoon graveyard.
It isn't Euclidean. It is Minkowskian.
Spacetime, not space, is 4 dimension and the geometry isn't euclidean. Flat spacetime is set in the framework of a minkowski space (this is more akin to a hyperbolic than euclidean geometry). A geometry is only Euclidean if it follows the five axioms as laid out in Euclid. In the geometry of spacetime the parallel postulate is not held (this also means as a consequence the pythagorean theorem and euclidean distance formulas do not hold), and is hence a non-Euclidean geometry. Furthermore space (not space-time) in a flat space-time is a 3 dimensional euclidean space as represented clearly in the metric using a (-+++) signature where ds^2=-c^2t^2+dx^2+dy^2+dz^2 -> ds^2 = dEuclidean^2-c^2dt^2. Furthermore 4-D Euclidean geometry would present a signature of ++++ (or equivalently all +1 along the diagonal inside of the metric tensor), is incompatible with all relativistic effects: time dilation, speed limit of c, and etc all require that the spatial and time components be of different sign. If you prefer to discuss this from finitist perspective replace all differentials with deltas and the arguments still hold.
@@mattsgamingstuff5867 Depends on what tools you possess ie what Coven you practice your witchcraft.
In the end y'all will come back to intuitionism.
Theorems are the exact opposite of theory.
They should be called factems.
Euclid is divine, the decoding of God's design.
GOD HAS NOTHING TO DO WITH THIS.
@@rogerstone4948 Nothing else makes sense to me.
@@rogerstone4948 seethe.
Here we go again round 2 of an unnecessary and irrelevent attack on Euclid's 'Elements' by a modernist who presumably thinks he could have done it better, an attempt at the restortion of a masterpiece of geometry that isn't damaged.
[1] Norman you are not the first mathematician to question whether the arcs intersect, a visiting mathematician at Gresham College did also and gave the mean value theorem as a solution. Unnecessary! All we need to appreciate is that if we construct the square of the base i.e. the square of the radius, then the two arcs will terminate in the corners opposite those that they originated in and MUST THEREFORE INTERSECT. Nothing is added to the proof by embellishment.
[2] Considering that all they had back then was a stick & sand box or a stylus & clay tile or ink & parchment on which to draw their geometrical figures, Euclid gave his BRILLIANT DEFINITIONS OF THE PIONT & LINE to stop super bright students including them in the resulting proofs. Looks like he failed as they are still debating them today as somehow having a negative influence on the proofs. WHAT RUBBISH! Euclid's points were rehashed by Richard Dedekind in the 1800's as cuts on the number line and with the advent of visual computer proofs the super bright things will be debating what definition is appropriate for the 'pixelated point and line' for decades yet to come.
[3] To help you understand Euclid Norman, lets take 4 SQUARE absolutely identical engineering gauge blocks and assemble the 4 to make a bigger square. The dimension of the square will be double that of a single block and the 'lines' seperating them will form a cross and wont affect the dimensions in any way what so ever yet will still be apparent. The line is not a line but an interface of contact. Like wise where the two interfaces intersect in the middle of the square is just Euclids 'point' which will be evident to us but have no physicality. As I have already said his definitions of the line and point are brilliant. 10/10 Euclid. 0/10 Norman.
[4] May I suggest Norman that you might be more gainfully employed revisiting the prime factorisation of the composites starting with the composite of two primes. Maybe you will then demonstrate as I have found after a critical anaysis that there are no shortcuts and no one method shows any advantage over another except simplicity, so it's back to good old fashioned repeated division preferably on a computer with a reduced number field. You should also arrive at the conclusion that manual factorisation is futile and therefore a complete waste of anyones time and should only be carried out by a computer. As for all the factoring sieve methods one reads about but cannot understand, perhaps they should hold 'Factoring Olympiads' to publicly demonstrate which one's works best and their limitations. Enjoy!
NJW----you are doing just fine! I noticed in my work life that everybody standing around thought they could do a better job than the guy actually doing it! LOL
@@acudoc1949 Euclid ---- you did absolutely great man and I honour you. I noticed in my work life that everybody standing around thought they could do a better job than the guy actually doing it! I gave the reply to my comment a thumbs' up.