According to Karl Jaspers, evil is the so called “own opinion”, evil is absolute selfishness: it ignores the interests of others. In essence, I agree with this. And from this it also follows that evil is an own goal: the environment - society – punishes the absolute selfishness, because it threatens the members of society and so itself¹. Thus, this approach also connects evil to the ignorance in one’s own opinion, the worng vision of past and future. This means that the evil commits its evil acts on the ground of incorrectly interpreted motives, it is unaware of the future consequences of its actions.

__________________________________

¹ The ‘goodness’ of the existence in society lies in this.

Abstract: If you define the existent as what exists is what "has an effect", then the potentially existent also exists. The in the present - at different points in space – currently existents have no effect on each other because any effect known to us spreads by an infinite velocity, i. e. time is needed until an effect gets from one point of space to another. ... Mathematics deals with potential and actual existent as well. The set of natural numbers is potentially infinite. Cantor's notion of an infinite number turned the existence of an infinite number to an actual existent. This infinite number is not constructed from an existing number, but he said that it is beyond all existing natural numbers, it is transfinite. ...

After registration, you may ask for a translation of the full text by email. 

Abstract: I try to approach the problems of philosophy by the means of mathemathics. Thus, a conflict of faith and knowledge, or rather their unity will be approached by a mathematical metaphor. Knowledge is a point in the information field, as I imagine, and belief is a vector. The greater the likelihood of faith, the longer vector, I think, and the greater the potency. The vector has a direction, just as faith does which gives direction to our actions. More precisely, faith based on knowledge governs our actions. In bold: the material points correspond to the knowledge points, and the belief system, or motivational system, is the equivalent to the field force. ...

It is very revealing as Poincaré distinguishes between mathematical and physical continuity¹, but it is even more interesting how he views the continuum. He introduces the concept of different orders of continuity; first-order continuity is what is provided by adding rational numbers to the number line, and second-order continuity is, which appears when irrational munbers also appear on the number line.²

__________________________________________________________

¹ As he described in his writing titled "Science and hypothesis", which is more an essay of philosophy of science, rather than a mathematical writing.

² Quoting Poincaré: "For brevity let me call the set of members which form under the same laws as the sequence of rational numbers as the first-order mathematical continuity. If we insert now new members according to the laws of forming the incommensurable numbers, the resulting set hereinafter will be called the second-order continuity." (This is not a translation from the original language but from Hungarian language.)

After registration, you may ask for a translation of the full text by email.

I begin with a quote, because I could not formulate better my starter thoughts of this writing. The quote is from the early 60s, but today it is still up-to-date.

"... The whole of modern mathematics is essentially based on the concept (or idea) of the intense actual infinity. In set theory there is much less developed extensive idea of infinity. Cantor's predecessor, Bolzano, whose views have greatly influenced the formation of set theory, trying to find a solution to the infinite quantity (extensive infinity) problem. However, the set theory was formed as a very rich theory of intensive infinity, while the extensive concept of infinity was essentially a stranger."¹

Cantor's concept of infinity is about extensive infinity, but he did not link this concept with intensive infinity. There is no connection between extensive and intensive infinity although there is an operation: the reciprocal function, which could lead from one to the other. The reason for the lack of resolution is "the gap between the continuum and discreteness" as it is called by A.A. Fraenkel and J. Bar-Hillel².

____________________________________________________

¹G.I.Naan, Infinity and the Universe, „Idea of infinity in mathematics and cosmology" (The quote is not the original text, it is translated from Hungarian.)

²A.A. Fraenkel , J. Bar-Hillel, Foundations of Set Theory

After registration, you may ask for a translation of the full text by email.

In the item titled "Two-element numbers and geometry” I mentioned that two-element numbers are closely related to infinity. Now, I should add that they also play an important role in different formulations of the (independent) continuum hypothesis, so in the descriptions of different mathematics.

Cantor defined infinite numbers that are located beyond natural numbers. These are the transfinite numbers.

I have got my idea for a new interpretation of infinite numbers from computer plotting of negative numbers. This method is similar to the usual decimal representation of fractions in the system of real numbers. For example, a special infinite number can be written as the following:

...999

So in the case of the ...999 number there is an infinite sequence of digits, ie. countably many 9 integers represent a special infinite number.

Apparently I did not do anything else than the real-digit representation of fractions. What can we say about this infinite number? First of all, the same as about the real -1, ie.

…999 ²=1                                *

After registration, you may ask for a translation of the full text by email.

I discovered two-element numbers for myself in the second half of 70’s. At that time it was difficult to obtain information about them, because there was no internet. I talked about it at the university ELTE, and I received material on Clifford and Cayley numbers. The two-element numbers, however, were simple, but they were as great and useful as complex numbers. By the way, one of them was just the complex number system. The other one I called hyperbolic numbers, since multiplying the numbers in this system we got the Lorentz transformation, ie, the hyperbolic rotation. The third most simple numbers were the parabolic numbers as I called them, in their system the picture of a product was an offset along a straight line. Thus, by multiplying numbers we got the three major movements: translation, rotation and hyperbolic rotation, ie, the Lorentz transformation.

After registration, you may ask for a translation of the full text by email. 

By two-element numbers I mean numbers which can be written in

a + bw

form where 'a' and 'b' are real numbers and w is a special number with following properties:

w² =- 1 or 0 or +1

and when w² = 1, w is not the real ± 1 number. In the literature these are called as complex numbers, kinds of Study numbers or dual numbers and hyperbolic numbers or perplex numbers.

In the item titled "Comparison of two-element numbers" I have already written about these numbers, their sum is represented as 'vector sum'. I also wrote about here that triangle inequality of these numbers is the same as in the spherical plane or the Euclidean plane or the hyperbolic plane.

After registration, you may ask for a translation of the full text by email. 

Content: Two-element numbers have three basic systems, comparison of the properties of the complex, hyperbolic and parabolic (also known as Study's) number systems, basic arithmetic and geometric operations, a comparison of these, introduction of general exponential form, interpretation of trigonometric functions and hyperbolic functions on these number systems. (Literature: Catoni, other sources in http://arxiv.org )

 "A common experience, when some colleague would try to explain some piece of mathematics to me, would be that I should listen attentively, but almost totally uncomprehending of the logical connections between one set of words and the next. However, some guessed image would form in my mind as to the ideas that he was trying to convey-formed entirely on my own terms and seemingly with very little connection with the mental images that had been the basis of my colleague's own understanding-and I would reply. Rather to my astonishment, my own remarks would usually be accepted as appropriate, and the conversation would proceed to and fro in this way. It would be clear, at the end of it, that some genuine and positive communication had taken place. Yet the actual sentences that each one of us would utter seemed only very infrequently to be actually understood!" (Roger Penrose, The emperor's new mind)

"With the formulation of analytical geometry and the theory of algebraic functions, with the development by Newton and Leibniz of calculus, mathematics ceases to be a dependent notation, an instru¬ment of the empirical. It becomes a fantastically rich, complex, and dynamic language. And the history of that language is one of progressive untranslatability."(George Steiner, The retreat from the word)

After registration, you may ask for a translation of the full text by email.