I would go beyond that and claim a third conclusion: what we call knowledge has changed with time. To explain what I mean, let me discuss the example of Isaac Newton. Looking back at Newton's work, we can readily see which part of it we want to call science: his calculus, mechanics, and optics had tremendous later developments. His alchemy and his study of prophecies by contrast did not lead anywhere.
As to the esoteric use of the Scriptures to understand history, it continues to this day, but most scientists know that this is nonsense and this opinion is supported by statistical studies.
A modern scientist distinguishes readily between Newton's good science and his pseudoscientific endeavoJs. How is it that the same admirable mind that unveiled the secrets of celestial mechanics could completely go astray in other domains? But what track? The progress of science is not just that we have learned the solution of many problems but, perhaps more important, that we have changed the way we approach new problems.
We have thus gained new insight into what are good and bad questions and what are good and bad approaches to them. This change in perspective is a change in the nature of what we call knowledge. And this change of perspective gives a contemporary scientist, or an educated layman, some intellectual superiority over giants like Newton. By intellectual superiority I mean not just more knowledge and better methods but in fact a deeper grasp of the nature of things.
In this chapter the examples will be easy, but the reader should be warned against the natural tendency to accelerate through what appears to be technical stuff.
On the contrary one should slow down! So, here we go. This means that the length of the edge AB is the same as the length of the edge A'B'. What this means is that if the two triangles are drawn on paper and you cut them out with scissors, you can move them around and superpose them exactly.
You may have to flip one of the two triangles recto-verso before putting it on top of the other. Using the pieces of paper you can also make clear what you mean by equal edges they can be superposed exactly or equal angles same thing. Why then did people ever get excited about "theorems of geometry" such as the one we just discussed?
Actually, a proof of this result can be obtained by staring at the following figure: b a. The Pythagorean theorem is useful knowledge. It allows us, for instance, to produce a right angle if we have a piece of string. Here is how. We make marks on the string so that it is divided into twelve intervals of equal length we may call this length a cubit.
Then we use our string to make a triangle with sides 3, 4, and 5 cubits: the angle between the sides of length 3 and 4 cubits will be a right angle. The ancient Greeks loved arguing, and they loved geometry because it gave them a chance to argue and to come to indisputable conclusions. Geometry, as Plato noted, is a matter of knowledge, not just of opinion. In Book VII of the Republic, he places geometry among the required studies for the philosophers who are going to rule his ideal city.
In a very modern discussion, Plato remarks that geometry is practically useful but that the real importance of the subject lies elsewhere: "Geometry is knowledge of what always is. It draws the soul towards truth, and produces philosophical thought. Plato refers here to geometry in the plane and notes II.
But the remarkable thing is that modern mathematics is done precisely in the way that Euclid presented geometry. Let me say this again. There is some flexibility in selecting the rules of deduction, and many choices of axioms are possible. Once these have been decided you have all you need to do mathematics.
One terrible thing that could happen to you is if you reach a contradiction, that is, if you prove that some statement is both true and false. This is a serious concern because Kurt Godel5 has shown that it is not possible in interesting cases to prove that a system of axioms does not lead to contradictions.
What I am trying to say is that most mathematicians are not distracted by Godel from their usual routine: they don't expect a contradiction to pop up in their work. Mathematics as done by mathematicians is not just heaping up statements logically deduced from the axioms. Most such statements are rubbish, even if perfectly correct. But why should that be the case?
And, in fact, what does it mean? These are difficult questions. We shall look at them in the next chapter and later. Before that it is desirable to have a look at the role of language in mathematics. Sort of. Of course, there are grammatical rules, but they are so messy and fuzzy that translation by computer from one natural language to another is a difficult problem.
That would be quite disastrous. The way out of this difficulty is to show that in principle we can dispense with a natural language like English. In other words one can in principle give a completely formalized presentation of mathematics. Why only in principle and not also in fact? One argues quite convincingly that the formalized text could be written, but this is not done.
Indeed, for interesting mathematics the formalized text would be excessively long, and also it would be quite unintelligible by a human mathematician. There are a few tricks that make life simpler. One may also introduce abuses of language: some controlled sloppiness that won't lead to trouble.
But for an ordinary mathematical text, one has to depend on the somewhat fallible intelligence of a human mathematician. In the best cases, the style is clear, elegant, beautiful. Most mathematicians would say that good mathematics consists of those statements that are interesting, that good mathematics has meaning, and that it is organized in natural structures.
To discuss the views of Felix Klein we need to actually do some mathematics, in fact, some geometry. And the proper way to proceed would involve axioms, theorems, and proofs. But I do not want to assume that the reader has professional expertise in mathematics or wants to acquire it.
I shall therefore use the approach of the Greeks before they formalized geometry in the manner of Euclid's Elements. I shall ask you to stare at figures and make simple deductions or believe some statements I shall make.
Think of yourself as a lover of philosophy in ancient Athens. But you are not afraid. You enter. We say that the plane is the space of Euclidean geometry, and we also have a notion of congruence. The motion should be rigid, that is, it should not change distances between pairs of points. Rigid motions, i. Euclidean geometry is very natural to us, but we shall see that there are other interesting geometries in the plane.
If we want to keep the concepts of straight lines and parallel lines but not those of distances between points or value of angles, we obtain affine geometry. Here, besides rigid motions we also allow stretching and shortening distances. Instead of congruences we have affine transformations.
Note that a square, by a rigid motion, stays a square of the same size, oriented differently:. D Affine geometry of the plane is defined by a space-the plane-and by the affine transformations. Let us mention in passing that the notion of the middle point of a segment makes sense in affine geometry, even though the notion of the length of a segment does not make sense.
Another kind of geometry is projective geometry, which arises naturally from the study of perspective. Note that the parallel sides of the table are no longer parallel in the picture. In the picture a point at infinity becomes an ordinary point of the plane.
In projective geometry we have a space called the projective plane, which consists of the ordinary points of the plane and the points at infinity. If a figure is drawn on a table and you give a correct perspective rendering of this figure on a screen, you establish a projective transformation between the plane of the table and the plane of the screen. When a point P of the table is represented by a point P' on the screen, we may say that the projective transformation sends P to P'.
The midpoint of a segment is not a good concept for projective geometry, but the cross-ratio is. The numbers a, b, c, d may thus be positive, negative, or 0. The quantity A, B; C, D equal to c-a c-b c - a d - b.
It does not depend on the choice of 0 or what we called the right of 0 and the left of 0. Using A', B', C', D' would give the same result. While the ideas we have just discussed go beyond Plato, he could have understood them.
Plato might not have been happy with this next paragraph, and perhaps you won't be either. Read it anyway, but don't get stuck. One may think of complex numbers as points in the complex plane. We define the complex projective line to consist of the points of the complex plane and a single extra point at infinity. Specifically, the point a complex number z is sent i.
The complex projective transformations transform circles into circles with the understanding that a straight line plus the point at infinity is considered to be a circle. In other words, complex projective transformations preserve the cross-ratio. Using the definitions I have just given it is a simple calculation. Let us now step back and see what we have. In the cases we discussed, the space is a plane with points at infinity possibly added.
But the plane was just for easy visualization; other spaces for example, three-dimensional space could be used. In mathematical parlance, the words space and set are more or less equivalent, meaning a collection of "points" in the case of a space or "elements" in the case of a set. The idea of Felix Klein is that a space and a collection of transformations defjne a geometry.
Introducing different geometries allows us to put some order in theorems. Then there is a straight line through Q, R, S. What kind of geometry does this correspond to? There are straight lines, no parallels, no circles. So, it is a good guess that Pappus's theorem belongs to projective geometry. Perhaps it is at this point that you start feeling there is more to geometry than a legalistic certification of theorems.
There are ideas-ideas that Plato could understand. So, we have seen that Pappus's theorem belongs to projective geometry. Classification is a great source of satisfaction for scientists in general and mathematicians in particular. For a problem of projective geometry, you will use another bag of tools containing projective transformations and the fact that they preserve cross-ratios. A problem may be fairly easy if you use the right bag of tricks and become quite hard if you use the wrong one.
To convince you that the Erlangen program is a useful piece of mathematical ideology, I would like now to discuss a diffic ult problem. Here it is:. Claim: M is the midpoint of the segment uv.
Now let me explain that a professional mathematician would not call this a very hard problem. In fact, it looks like an easy question of elementary Euclidean geometry.
One immediately notices that the angles at 5 and Q are equal. Then one tries to use standard results about congruent triangles like the one in chapter 2.
Then one may start having doubts. Is it really true that M is the middle of UV? In fact, yes, it is true. The reasonable thing to do in this sort of situation is to sleep on it. I am a most reasonable person, so that is exactly what I did after my colleague Han Vardi showed me the problem and I could not readily solve it. If you really want to crack the problem, you can now do either of two things. This method is due to Descartes. It is often long and inelegant, and some mathematicians will say that it teaches you nothing: you don't get a real understanding of the problem you have solved.
To most mathematicians this is the preferred method. Let me now briefly outline a proof o f the butterfly theorem. You may work out the details or be satisfied with the general idea, as you prefer. Consider the points A, B, P, R,. This shows that the midpoint of the segment UV is. These natural structures need not be easy to see.
The mathematician thus has access to the elegant world of natural structures, just as, in Plato's view, the philosopher can reach the luminous world of pure ideas.
Today's mathematicians are thus the rightful descendants of the philosopher-geometers of ancient Athens. They have access to the same world of pure forms, eternal and serene, and share its beauty with the Gods. And in some form or other it remains popular with many mathematicians.
Among other things it puts them above the level of common mortals. Mathematical Platonism can not be accepted uncritically, however, and we shall later dwell at length on this problem. But at this point a shocking question should be discussed: how did our butterfly theorem find itself included in a list of IIanti-Semitic problems"?
The setting of the story is the Soviet Union, and the time is the s and s. Scientists then were to some extent sheltered from the prejudices of the ruling caste.
In particular they limited the admission of Jews and certain other national minorities to major universities in particular, Moscow University. Proving the butterfly theorem is on the list of the "murderous problems," and you can see why: the natural way to attack the problem will probably lead you nowhere. Of course there is a relatively simple solution, and a seasoned mathematician will eventually find it.
But think of a young person who takes an examination to enter the university and has to crack such a problem in limited time. And, as one colleague remarked, "However tragic this problem of ethnic discrimination may have been in individual cases, it is only a marginal tragedy next to a much greater tragedy. But even if you want to consider it as a marginal issue, the use of mathematics to implement ethnic and political discrimination is very disturbing to mathematicians.
We thought of mathematics as living in a serene world of forms, beauty, pure ideas, and here it sits among other tools of repression. And some colleagues in the West seem eager to believe in this sudden mutation. How did I drift from mathematics to this particular political discussion?
The plight of other groups is currently a more pressing question than that of Soviet Jews. So, should I not leave political ugliness aside and concern myself rather with the beauty of the Platonist world of forms? Moral issues, one may say, are not part of science. We shall later meet other examples of this unfortunate situation.
But there are more things in mathematics, than just geometry. For example, arithmetic: we start with the numbers I, 2, 3, 4,. One can define primes, 2, 3, 5, 7,. The number 1t is the circumference of a circle of diameter 1; it is a modern result from the eighteenth century that 1t is not a fraction.
I might easily be carried away and start telling the saga of mathematics. How one proves a miraculous formula like2 1 1 But this is not my purpose here. What I have just said indicates two fundamental tendencies in the development of mathematics: diversification and unification. It is clear how diversification arises: everyone can set up a new system of axioms and start deriving theorems, creating a new branch of mathematics.
So it would seem that mathematics disintegrates in front of our eyes into a dust of unrelated subjects. But the subjects are not unrelated. For instance we have just seen how real numbers such as f2 or 1t appear in questions of geometry. In fact, there is a deep relation between Euclidean geometry and real numbers.
Between Euclid and the nineteenth century the proper way to handle real numbers was through geometry: a real number was represented as the ratio of the length of two line segments. This was at a time when the modern theory of real numbers had not yet been developed. This may come as a surprise after all the fuss we have made about defining mathematics in terms of axioms. From these facts one then proceeds to derive new results. Doing this repeatedly, one can hope to present the whole of mathematics as a unified construction based on only a small number of axioms.
This hope has been a central driving force for mathematics through the ninteenth and twentieth centuries. One can say that the hope has been fulfilled but not without crises and surprises. We shall have the occasion to come back to parts of the story later but will now pause to look at the curious case of the French mathematician Nicolas Bourbaki.
For historical reasons France is a strongly centralized country. As a consequence, scientific research and teaching have often been under the control of a few powerful old people. And this control has been painful for the young bright scientists. After a while, however, the old tyrants die.
The power is seized by the young bright scientists, who, in the mean time, have aged. After a while then they became old tyrants, and we are back to the old situation.
But sometimes the results are excellent. Why excellent? Because it frees the bright young people from time-consuming responsibilities and allows them to be scientifically productive. We have seen that for similar reasons Soviet science was of high quality in certain areas of research. Analysis contains things like multiple integrals and Stokes formula, which are of everyday use in theoretical physics.
So the idea was to develop analysis in a fully rigorous manner, all the way to Stokes formula. Constructing analysis on a rigorous basis means we start from axioms. But not axioms of analysis! As we have seen, we want to build all of mathematics, including analysis, in a unified way from just a few axioms. And one century of mathematical mind searching has shown that a good idea is to start with the axioms of set theory.
This looks tremendously uninteresting and unpromising. In set theory you can analyze axioms and logical rules of deduction with the greatest clarity. How do you obtain the rest of mathematics from set theory? By counting the objects in a set, you get the integers 0, 1, 2, 3,. From the integers you can define fractions and real numbers using ideas of Richard Dedekind or Cantor. And so on. Even though not everybody liked it! Of course the mathematicians who launched Bourbaki in the mid-'30s did not know where their enterprise would lead.
But they were enthusiastic. And very sharp. Take for instance Andre Weil. To someone who said, "May I ask a stupid question? Eventually he went to the United States, where he pursued a brilliant mathematical career, which eventually led to the Weil conjectures. The later proof of these conjectures by Grothendiek6 and Pierre Deligne7 was an important moment in twentieth century mathematics. Of course the political judgment of Andre Weil with respect to World War II may be questioned, but one must recognize the independence of mind that it shows.
And this independence of mind served him well in his creative mathematical work, which shows a healthy disrespect for the achievements of his predecessors in the field of algebraic geometry. Speaking of Andre Weil, we must mention his sister, Simone, who was the better known member of the Weil family in the European intellectual community.
She wrote a number of influential books on her social and religious experiences. The prewar problems and wartime horrors hurt her deeply, and she died in in England from self-inflicted starvation. And what about Bourbaki? The last two publications date from and , and there will probably be no more. Bourbaki is dead. To go beyond these two or three names would be difficult, and different mathematicians might end up with rather different lists.
We are still too close to the twentieth century to have a satisfactory perspective. In other cases the work of a mathematician turns out in retrospect to have changed the course of mathematics, and his name will emerge as one of the very great names of science. One name that is certainly not fading at this time is that of Alexander Crothendieck. He excluded himself. How this mutual rejection took place I shall later relate. Crothendieck started his career with problems in analysis, where he made contributions of lasting significance.
But the great labor of his life was in algebraic geometry. The curves called conics or conic sections were studied by later Greek geometers and include ellipses, hyperbolas, and parabolas. Take the following geometric fact: through five given points in the plane, there passes in general just one conic.
The more precise theorem is the following: if two conics have five points in common, then they have infinitely many points in common. This is what has happened in algebraic geometry: the way to develop the subject has been told to mathematicians, so to say, by the subject itself. For that reason, classical algebraic geometry largely uses complex rather than real numbers. This means that besides the real points of a curve one also considers complex points, and it is natural to introduce also points at infinity.
Of course you will want to study not just curves in the plane but also curves and surfaces in three-dimensional space, and go to spaces of dimension greater than three. The way we just introduced them, algebraic varieties are defined by equations in the plane or some space of higher dimension. But it is possible to forget about the ambient space and to study varieties without reference to what is around them.
This line of thinking was started by Riemann in the nineteenth century, and it led him to an intrinsic theory of complex algebraic curves. The study of algebraic varieties is what algebraic geometry is about. It is a difficult and technical subject, yet it is possible to sketch in a general way how this subject has developed. One can add, subtract, multiply, and divide real numbers in the usual way dividing by 0 is not allowed , and this is expressed by saying that the real numbers form a field: the real field.
Similarly the complex numbers form the complex field. And there are many other fields, some of which called finite fields have only finitely many elements. But why go from real or complex numbers to an arbitrary field? Why this compulsion to generalize? It is just as simple, and being more general, it is also more usefuL Stating things at the proper level of generality is an art. At this point I would like to jump from algebraic geometry to something apparently quite different: arithmetic.
Fermat's last theorem is the assertion that. In , Pierre Fermafl thought that he had a proof of this assertion later called Fermat's last theorem , but he was probably mistaken. A true proof was finally published in by Andrew Wiles. Presented like this, arithmetic appears very similar to algebraic geometry: one tries to solve polynomial equations in terms of integers instead of complex numbers.
Can one then unify algebraic geometry and arithmetic? Actually there are deep differences between the two subjects because the properties of integers are very different from those of complex numbers. This fact is known as the fundamental theorem of algebra. It is flat and is barely noticeable when carrying it. I carry this gun everyday and sometimes forget it is there.
This is a double action pistol, while some prefer the constant pull of striker fired guns, I prefer the heavy first shot and lighter follow ups. In my hands follow up shots are quick and easy. Which is what I practice most for self defense.
The manual and magazine safeties were also a consideration for me. When storing the gun for the night I put the manual safety on, but carry with it off. I have heard a lot of opinions on magazine safeties and understand both sides of the argument.
Some prefer not to have it due to tactical reloads and being able to fire the gun with one in the chamber and magazine removed. I prefer the magazine disconnect. If you got into a fight and someone was trying to get your gun then all you have to do is drop the magazine and render the gun unsafe. I also use this function when I want to keep the gun ready but am worried about someone finding it.
With the mag removed and separate from the gun then there are no worries. Some are still available through LE contracts. Upon first look at the Extreme Violence mod, my eyes lit up. I was in heaven! A mod where I could murder other Sims, as I had been talking about how great it would be to be able to murder in the game.
It adds a specific challenge, especially if you turn on autonomous killing. I was shocked to find that after my Sim killed three people in a row autonomously on her own without me telling her to , the Grim Reaper had had enough and was beating her up.
You can read about all the details HERE. After adding the Lot Trait the default Amount of 10 Simoleons will be activated. This mod adds new insult and argument social interactions for your Sims.
All interactions are set to. As this is a script mod, you must enable script mods in your game options, and both files must be nested no more than one folder level inside your Mods folder.
Though sims 3 showed a glimpse of a little boy being bullied in a trailer and over coming that. Twallan made a story progression mod that allowed this kind of scenario for people who wanted it in game.
I feel that sims is therapeutic for people in general. You can have your Sim do deadly interactions, or they can do non-deadly interactions where their victim has a chance of surviving.
There are a few different options for killing Sims. You can shoot Sims, stab them in the gut, or stab them in the chest. Some interactions, such as the stabbing, will leave your Sim bloody and needing to be cleaned up, while others, such as ranged shooting, will not leave your Sim bloody but will leave a pool of blood on the floor.
One downside of using this mod is that other Sims will run away from your Sim and be scared when they see your Sim brutally murder another. The interactions are graphic, as they do involve blood and gore; they leave a hole in the stomach or chest where you stabbing occurred. There are deadly interactions with a chance for survival, though either way, you might want to be careful when turning on autonomous killing, since your Sim may kill off their own family members.
Messing with these mods tends to make me laugh, and they can be very therapeutic after a hard day. My best guess is that it would allow a vampire to kill by draining all their blood, or something along those lines.
With the Extreme Violence mod, Sims can have a reputation. The Sim reputation will determine how another Sim reacts to them and how they behave in their presence. You can sabotage a reputation by accusing a Sim of being a public offender, a murderer, or just being violent. Even if none of the above is true, you can still make up these rumors to make sure that this other Sim is feared for whatever reason.
There are plenty of self-interactions, too, such as taking different kinds of selfies. One of my favorite selfie interactions has the Sim flipping the middle finger. The Extreme Violence mod will allow you to do more devious things with your partners in crime.
The mod has many other features that it would be too much for one article. What are your thoughts? Let us know! Join us on Discord, on our Facebook page, or Twitter. Whereas other age groups get stuck behind with limited interactions. So any mod that gives Toddlers, Children, Teens, and Elders a chance to shine is an automatic win in my book. Thankfully, most of these school mods do exactly that: especially for Toddlers to Teens.
Check out these incredible and incredibly fun school mods to actually make learning fun again. This closes off some interactions for them, but it definitely opens up more in terms of a career. Mahabharatham - One of the great Indian Epic to know each and every Indian and each and every one in world too. For Old Series: Please open this link http. Download it from a site like YouTube using software such as 4K Video downloader. One can configure the Smart Mode to download multiple files at one go.
Vijay tv mahabharatham full episodes tamilrockers online. Details of Mahabharatham. Episodes Full HD print from 1 to Excellent HD print - 5. Quick Note: You should ideally just need to download and install the files available. And instead of sending our Sims down the ever-famous rabbit hole, it allows us to follow our Sims around and control the flow of their workday.
Best part? Everything in the lot is interactive and made of vanilla base game items. The biggest differences are 1 it also gives Children the option to stay out of school, and 2 it leaves Children and Teen Sims alone from the get-go. They can basically hang around the house as they please — with no fear of repercussion or penalty —until their birthday.
Should that happen, all you need to do is use the cheats careers. Essentially the really hefty, extensive get-out-of-school mod and the simple, straightforward get-out-of-school mod.
It disables the auto-enroll feature by default. Which means your Toddler or Child will not automatically enroll in school once they age up. You have to manually do it. This mod with a lengthy name has a couple built-in failsafes and two variations, too. Autocad free download with crack. Whichever of the three you download is definitely a matter of preference and little else.
I personally love the concept of this mod, no matter how simple it seems. And the creator definitely delivered. The premise is that the actual time your kid Sims get to leave school is based on their grades.
Kinda like real life. The default time that all school children come home to is 3PM. A students get to come home at 1PM for exemplary performance, probably. This was created to address the issue of there just being not enough time for Teens.
Simply enable cheats by typing testingcheats true into the command bar and then click on the Teen while holding down the Shift key.
This should bring up the cheats menu. But instead of a mod that rewrites the whole educational system, this introduces a new Event that your parent Sim can plan. Your school will have to have a kitchen if you want the staff to cook, and a microphone if you want the teachers to teach.
Yes, you can actually have a school cafeteria for your Student Sims to grab some lunch! The mod is actually a bunch of her other education-related mods all bundled together. And a different mod even adds an Online Schooling option to the game! Just plop your Sim in front of a laptop or computer. A new interaction that reads Attend School Online should pop up.
Be sure to read it carefully to optimize the experience! Now this mod adds a new education system, and a new social event that may bring back fond memories for people who played The Sims 2. The event begins once the Headmaster enters your home. You need to impress them via successful social interactions, lots of boasting, and a game of chess. Pretty pricey in-game, but hey.
Lua 14 min ago 1. Prolog 29 min ago 2. Java 37 min ago 2. Python 52 min ago JavaScript 56 min ago 1. Python 1 hour ago Java 1 hour ago 1. Can you become the next Grand Slam Champion?
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