Hrvoje1

test

[math]\sqrt{a}[/math]

[math]t_1=\frac{1+(n-1)(1-v/c)}{c/d+(v/d)(n-1)(1-v/c)}[/math]

So, the question is, why are random sequences in reality, in nature, not like that, describable by simple rules? With a perfectly balanced dice, getting every time the same number would be in conflict with the law of large numbers. I think I am at the end of my understanding am I confronted here with some oddness or not. My intuition tells me that random number sequence cannot be generated by a simple rule, because, it is not generated by any rule, by definition, and that actually tells me my logic too. Maybe you were seeking to much for some mystery here, when there is none, studiot? Although, that is also some kind of rule (no rule). And it's a simple one. And if you draw a random line in a coordinate system, with a free hand, that is, without a ruler and compass, I bet it would be asymmetric, for the same reason.

I am sorry, but I don't get it again. If that sequence would reach a very large number n, and continue consistently on for as long as we can follow, according to the same rule, as an output of a random number generator, I wouldn't call it perfect, but extremely poorly designed random number generator. In fact I don't know what could be worse than that. Maybe to produce a dice for a sole reason of using it for random number generation, and whenever you throw it, you get the same number? Or a random number generator that produces such output: {1,2,3,4,5, ...}? If I am not missing something here, that would be the same class of failure. Right? I mean, if you construct something that is not supposed to output something that can be described by such a simple rule, and you still get it, then you didn't do a good job. OK, I got it you are right, there is no reason why should we expect that random sequences cannot turn out to be easily described by simple rules. Especially if we believe that nature is regular in its essence.

Here you go again. If I said that symmetry appears to me as an example of regularity, then it means that I don't think these words are synonyms. ... in a sense that regularity can have other meanings. They don't cover each other 100%.

So if you start with symmetric functions alone, the only way to get an asymmetric one ... that's actually cool.

OK, now you are avoiding the issue, you realized that I believe in Wigner's unreasonable effectiveness of mathematics in natural sciences, and now you are after me. If you draw something on a sheet of paper, the trace of the pencil is "in nature", and the idea of what it represents was at first in your mind, and it can be in someone other's mind if it is shown to someone else, and although minds are natural, their content can be real or imaginary. Regularities that are in our minds are in connection with regularities in nature, as long as our minds are in connection with reality, that I consider nature. But that is not exactly the question I raised. I never said that symmetry and regularity have identical meaning, you were the one that was pushing the discussion in that direction, explaining the differences, although, no one actually opposed. My drive to start the discussion was this insight: Any linear combination of even functions is even, and they form a vector space over the reals. Similarly, any linear combination of odd functions is odd, and the odd functions also form a vector space over the reals. The vector space of all real functions is the direct sum of the subspaces of even and odd functions. So if you start with symmetric functions alone, the way to get an asymmetric one is a linear combination of both an even and an odd component, and what is cool, is that you can achieve that for every possible asymmetric function. In that sense, symmetric functions are more fundamental then asymmetric ones. Right? It's kind of like asymmetry is an emergent property that none of symmetric components has. Although, it is a direct consequence of properties of both components, achieved by addition, so it doesn't quite conform to the definition, so let's drop the emergence digression, it was sufficiently discussed in the other thread.

I don't get it, does this "argument" lead somewhere? If this is true then f(x) might be A(x)sin(ω(x)), and feven(x) and fodd(x) are as previously described. So what is more fundamental principle in nature, symmetry or asymmetry? What is more fundamentally existing in nature, regularities or irregularities? If any of them was derivable from the other, I would be able to establish such relationship between them. What is 'more fundamental', is pretty much always relevant in science.

I think it can be decomposed as a sum of two symmetric functions, one even and one odd.

There seems to be no accidents, everything is somehow connected, for example, I started to read that text about groups in physics motivated by wtf’s response here So, if there are laws, are the laws of computation the most fundamental laws?

Any way you put it, it sounds as an oxymoron, somehow. I came up with the most interesting definition of regularity for this discussion: appearance or behaviour governed by an exact set of rules or regulations Of course that point symmetry requires that also, that was an instant blackout from my side. ... in fact, this was already at least equally interesting definition of regularity, or maybe even more interesting. So basically, are there laws in nature, or not.

At least axial symmetry requires that, ...

Irregular pattern is an oxymoron, a bit, if we assume regularity is needed to establish patterns.

I thought to leave it open to discussion, and gave examples of what my intuition tells me the regularity is. That may or may not be identical to established mathematical terminology or to your idea of regularity. I buy medium size, and I guess that could be another interpretation of regularity. I also think that everyone intuitively understands that symmetry of polygons requires that lengths of at least some of its sides are identical, and as such is related to their regularity, that requires all of their sides to be equal.

I came up with the question reading this article: https://mysite.du.edu/~jcalvert/phys/groups.htm , but such and similar questions were raised and reflected upon thousands of times, books were written on similar topics https://www.cambridge.org/hr/academic/subjects/physics/general-and-classical-physics/symmetries-physics-philosophical-reflections?format=HB&isbn=9780521821377 , so I am afraid I didn't do a good job.

It meant that I understand the definition of regularity with respect to polygons, which is binary. They are either regular (equiangular and equilateral), or they are not. However, if we change that definition, and admit some degree of regularity to cases that are somewhere in between two extremes, strictly regular on one side and totally irregular (in a sense that they have no equal sides) on the other, then symmetry would be an indicator and measure of such weakened regularity, maybe even the same thing, so that it might seem redundant to introduce the notion of "loose regularity" when there is already a notion of symmetry. Anyway, in case of triangles, by that definition, isosceles triangle (defined as the one that has exactly two sides of equal length) would be such case, more regular than totally irregular, and less regular than strictly regular. Since it has one axis of symmetry, while strictly regular have three, and totally irregular have none. Right? I think that if the question is pretentious, ignorant, vacuous, such that a mathematician would gladly leave it to philosophers to answer it, then it is a negative trait, otherwise, if it has some merit, for science or even for mathematics itself, then it is a positive trait.

True dat, but, for example, regular polygons are more symmetrical than those which are not, so I figured it would not be that wrong to talk about symmetry considering it as an indicator of regularity. And all other cases in which mathematicians use the term "regular" sound interesting, about some of them even I have a clue, but I don't want to talk about it. I would like to discuss is the ability to pose a philosophical question, as a result of reading a mathematical text, a positive or a negative trait?

True, but while this is trivial for mathematicians, it may not be for philosophers. First of all, one should show that this is actually true for any f(x): f(x) = [f(x)+f(-x)]/2 + [f(x)-f(-x)]/2 = f(x)/2+f(-x)/2 + f(x)/2-f(-x)/2 = 2*f(x)/2 = f(x) Then, one should prove that this actually represents a decomposition to an even and odd function: f(x) = [f(x)+f(-x)]/2 + [f(x)-f(-x)]/2 = feven(x) + fodd(x) feven(x)=[f(x)+f(-x)]/2 => feven(-x)=[f(-x)+f(-(-x))]/2=[f(-x)+f(x)]/2= [f(x)+f(-x)]/2=feven(x) fodd(x) =[f(x)-f(-x)]/2 => fodd(-x) =[f(-x)-f(-(-x))]/2=[f(-x)-f(x)]/2=-[f(x)-f(-x)]/2=-fodd(x) Finally, one can check that for a symmetric function f(x), one of such parts is zero. So for odd f(x): feven(x)=[f(x)+f(-x)]/2=[f(x)-f(x)]/2=0 For even f(x): fodd(x)=[f(x)-f(-x)]/2=[f(x)-f(x)]/2=0 If I wrote all of this in a philosophy forum, they would assume I am pretentious. I draw that line by mentioning something that appears to me as examples or regularity, such as symmetry and periodicity. How about you?

True, we should drop the emergence discussion, wtf, have you watched this? Godel Escher Bach Is the self-reference the essence of consciousness? Here it explains the difference between consciousness and sentience, which is slight, according to this: https://www.animal-ethics.org/sentience-section/introduction-to-sentience/problem-consciousness/ and it theorizes about central nervous system as a prerequisite for consciousness in living beings.

Let me tell you once again that I agree with you and get your points, I was just elaborating on them. Not every whole is something more than the sum of its parts. If its wholeness can be exactly described by addition of contributions of each of its parts, then these parts must be of the same kind, qualitatively, functionally, whatever, so that the whole can be only quantitatively different than its parts. If that's not the case, then the question is how many cases are there left, one or two? If you combine parts of different kind into a whole, then you usually get something that exhibits properties that none of its parts have on its own. If that is true for everything, as you say, then we have just one additional case. If not, if some additional complexity of the system (of the whole) is required for emergence to appear, entering some new level or dimension when combining parts, whatever you call it, then there are two cases. Or, if we are surprised by a result of combining parts into a whole, then it shows emergent properties to us, otherwise not. That is not very objective criterion for classification, but I didn't study it properly, as I already said, there surely are people who know the mathematical description of emergence better than me.

If you add 2 cm to 3 cm you get 5 cm, that is nothing more than sum of its parts, because it is exactly that, although it can be also a sum of its different parts, such as 1 cm and 4 cm, and I get your Feynman point about energy, applied to emergence. I can not tell how useful concept it is, emergence I mean, but I find adding an useful operation. Much more than these sophisms that I just demonstrated. There surely is mathematics to express emergent phenomena that does not require questioning of usefulness of addition.

Hey wtf, here are some attempts to answer the question: Mathematical Foundations of Consciousness AXIOMS AND TESTS FOR MINIMAL CONSCIOUSNESS What do you think about them?

Let's take symmetry of functions as an example of regularity, others may be their periodicity, etc. If one analyzes it, one can conclude that even functions, which are by definition those for which f(-x)=f(x), examples of these are polynomials consisting of even powers of x, and odd functions, for which f(-x)=-f(x), examples of these are polynomials consisting of odd powers of x, are actually exceptions, rather than a rule, ie that functions are generally speaking asymmetric objects with respect to the x=0 axis (or plane in 3D), that do not necessarily have anything to do with those that are symmetric (even and odd). However, the fact is quite the opposite, every asymmetric function can be represented as a sum of an even and odd part, like this: f(x) = [f(x)+f(-x)]/2 + [f(x)-f(-x)]/2 = f(even) + f(odd) So, for even functions, odd part equals to zero, and vice versa. That may be surprising, that such a simple logic shows the truth that may seem counterintuitive. Interesting is however, that symmetry in a microscopic world, for example in the world of elementary particles, is exact, while in a macroscopic world, for example in biology, it is only approximate. Why is it so? By that I mean that while hydrogen molecule is perfectly symmetrical consisting of two identical atoms, neither our bodies are perfectly symmetrical, nor we can produce any macroscopic object that is perfectly symmetrical. Is there a mathematical explanation for that fact, or does this question belong to a philosophy forum?

I think you are right, I believe that is a general idea on which that definiton from the paper is based on. Thank you wtf.