• I like sushi
    Don’t assume your singular brain cell is infinite just to make yourself feel better.
  • Wittgenstein

    I am a bit on the asperger side, but l can sense that you are kidding.
    Are you talking about this series.
    1/2 +1/4+1/8......till infinity =1
    This converges.
    Consider this series
    -1+1-1+1.......infinity, this is not well defined.
    So a bounded sequence can have a series that is not well defined.
  • Wittgenstein

    :wink: there is no end to a joke about infinity.See what l did there.
  • Shamshir
    No, I am not kidding.

    Here's an example - take a picture, 100 by 100 pixels.
    Infinitely zoom in - the picture is now infinitely large.
    Infinitely zoom out - the picture is now infinitely small.
    Each pixel within that picture is its own picture and has its own pixels.
    In the end - it all converges in to one; one picture, one pixel.
  • Mephist
    What about infinite lines in euclidean geometry? Are they allowed? Are they the same thing as segments of length one?
  • g0d
    Is there even such a thing as a strictly random number.Wittgenstein

    Perhaps we have no choice but to fix some meaning for 'random.' One fascinating way to do this is:

    Kolmogorov randomness defines a string (usually of bits) as being random if and only if it is shorter than any computer program that can produce that string. To make this precise, a universal computer (or universal Turing machine) must be specified, so that "program" means a program for this universal machine. A random string in this sense is "incompressible" in that it is impossible to "compress" the string into a program whose length is shorter than the length of the string itself. — Wiki
  • Shamshir
    They are. If you think otherwise, do please tell me why.
  • Mephist
    Which of the two questions do you answer? both?

    So, you say that infinite lines in euclidean geometry are the same thing as segments of length one, right?

    I thought this is obviously absurd. OK, if I have to say why: because infinite lines always intersect if they are not parallel, and finite segments can be not parallel and not intersect
  • Shamshir
    Which of the two questions do you answer? both?Mephist

    I thought this is obviously absurd. OK, if I have to say why: because infinite lines always intersect if they are not parallel, and finite segments can be not parallel and not intersectMephist
    Yeah, I got that, but don't see the problem with it - if you look at infinite lines as stretched out finite lines.
    You have two finite lines that are too short to intersect; you infinitely stretch them out and they intersect.
  • Mephist
    Yes, but we are speaking about measuring the length of segments. You can't stretch the segment if you want to measure it.
  • Shamshir
    Sure you can.
    To illustrate, you can infinitely divide a finite thing - which provides the notion it is an infinite thing, with infinite parts.

    Now let's bring up the issue at hand.
    Two finite things which are according to the aforementioned - likewise two infinite things, don't intersect.
    But the idea is that two infinite things must intersect.
    And my solution is simple - the halves of the one intersect in sum.
    They are finite and likewise infinite - according to the aforementioned, and intersect either way you look at them.

    And this comes about, due to it all always amounting to one.

    Sure enough, you could just draw two lines that don't intersect and showcase that, but the two non-interlockers along with their void still amount to one.

    Point being, if there is one, the lines can and cannot intersect, and all parts amount to one.
  • Mephist
    OK, so I can prove that 1 = 2. Here's the proof:

    I take a segment of length one and I stretch it until it becomes of length 2. But this is still the same segment, so 1 = 2. What's the difference between this and your argument? The halves of two are the same as one. Why is your argument not valid for finite lengths?

    I can even prove that a segment the same as a circle: just bend it and it becomes a circle!

    If I want to compare the size of two objects I can operate on the objects only with transformations that don't change their size: for example I can put them one on top of the other by moving them, but I can't split them in two or stretch them.

    I think I didn't understand completely your argument, but why can't you say the same thing for two instead of one?
  • Mephist
    if you don't allow the existence of infinite sets, you have to treat segments as a different kind of thing than a set of points.
    But in the galois field, they treat the segment made of points but don't use infinite sets, the one mentioned in the article.Can you send me any article,book recommendations that make your position clear to me cause l may be confusing your point here, I hope not.

    Sorry, I realize now that I didn't answer to this question.

    I read the article that you posted (https://plato.stanford.edu/entries/geometry-finitism/supplement.html).
    It's not true that "they treat the segment made of points". They say that "a point p corresponds to a couple (x,y)" and "A line corresponds to a triple (a,b,c)"; they don't say that "a line is a set of three points". But I agree that this is an example of finite models for Euclides' axioms, (by the way, in their original form Euclides' axioms are not expressed in a formal language, but I don't want to be picky on this point).

    The infinite model that I was referring to is the standard one, based on the standard topology of the real line (https://en.wikipedia.org/wiki/Real_line).
    I actually don't have a proof that there are no models of Euclides' axioms where a line is a finite set of points. On the contrary, I think you can easily build one by taking as space a finite 2-dimensional array of points, but this is obviously not the right model for the physical space.
    So, probably I should have said that: if you want to build a model of the physical space where a segment is described as a set of points (as in standard topology), you need infinite sets. Otherwise, you have to build a model of the physical space where a segment is not a set of points (an example of this is kind, not based on set theory, is smooth infinitesimal analysis: https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis).
  • Shamshir
    Quite. One halved is two, so by extension 1=2.

    What I've said is valid for finite lengths, but if it sounds incoherent it's due my inability to properly explain. I'm not sure how to remedy that.

    The best way I can put it is that one is the whole and thusly references all lengths without exception, and all lengths individually and as a contingent amount to one. The contingency of the whole amounting to repetition of one which derives distinctions precisely from the twists you mentioned.

    I'm really not sure how to put it better, ironically, as it all amounts to how you put it. :)
  • sime
    Suppose somebody sends you a circle in the post. You then proceed to verify that the circle isn't perfect. Does "perfect" here refer to an internal property of the circle or to our inspection of it?

    Here is what I think. Whenever we construct a set ourselves, we must choose the next element to include in the set, either explicitly, or by implicitly by defining a rule of selection. But in order to do so, it must be first be assumed that our elements are individuated a priori for a process of construction to make sense. Yet if we are given a set, say a parcel through the post, it's elements aren't individuated until we inspect the set. If the parcel we are given is called "infinite", all this means is that we shouldn't expect the termination of our parcel inspection to be decided by a property internal to the parcel.

    Mathematics tends to call parcels with non-individuated elements "equivalence classes" of elements. Like in the above example, this allows mathematics to either construct sets in a 'bottom up' fashion from elements, or to construct elements in a 'top down' fashion from parcels.
  • Mephist
    Well, I see that you are not speaking of mathematics or logic here. So I think I have not much to say in reply to your answer...

    But in my opinion the "infinity" in mathematics is not such an arcane entity as your "one" thing.
    Infinitely big and infinitely small are the "simplification" of extremely big and extremely small.
    So for example you can consider an infinitely long and infinitely thin line as a model of a very long and thin wire, because you want to "abstract away" the properties related to it's real length and width, and consider only the position and orientation of the wire.
    The only reason that infinite numbers were ruled out from mathematics (on the contrary of infinite geometric objects, that were always considered allowed) was the idea that you cannot reason about them without getting inconsistencies in logic. But now we know how to deal with infinity using formal logic in a perfectly sound way, and nearly all important mathematics is making use of infinite and infinitesimals.

    So, to me asking if infinite really exists is like asking if a plan really exists, since all "real" objects are 3-dimensional: it's only a mathematical approximation (simplification) due to the fact that you want to ignore one of the 3 dimensions. But probably this point of view is only due to my limited knowledge about philosophy..
  • alcontali
    owever, the statement, " an infinite number of elements " is a contradiction in terms as infinity is clearly not a numberWittgenstein

    In the viewpoint that math is about symbol manipulation formalisms, we may not even be interested in the concept "meaning" as some kind of correspondence factor with the real, physical world.

    For a starters, the symbol ∞ is just one character and not an ever-growing sequence of characters. Hence, it is perfectly suitable for participation in symbol-manipulation formalisms. An ever-growing sequence of characters, however, would be a problem, because in that case our symbol-manipulation algorithm may not even terminate.

    Secondly, the symbol ∞ is not irrelevant or immaterial, because it is definitely mentioned in usable reduction rules that can successfully extend classical arithmetic. For example: a + ∞ = ∞ and also a * ∞ = ∞, with for example, a ∈ ℝ. The symbol clearly has some kind of "absorbing" effect on other elements in its domain. You can actually obtain/generate the ∞ symbol by performing particular manipulations on more common elements of the domain, such as 1/0 = ∞.

    With the various reduction rules available, the symbol ∞ could actually be useful when you seek to produce a closed form output result from a particular input expression. The domain does not even need to be numerical and the algebraic structure not necessarily a field. The rule templates will undoubtedly still be consistent.

    For example, with a and b arrays, elements from the Array domain, and a + b defined in a particular meaningful way, and ∞ a meaningful extension to the Array domain, you will find that a + ∞ = ∞.

    So, it will still absorb the other elements during addition and multiplication. In that sense, it is a bit similar to the zero symbol, which also absorbs other elements after multiplication while leaving them unchanged after addition.

    In other words, a field or other algebraic structure can successfully be extended with the symbol ∞ while maintaining consistency and while satisfying the pattern in existing rule templates for the symbol. So, infinity may indeed not be a number but it is certainly a legitimate extension element in numerical algebraic structures.
  • g0d
    Secondly, the symbol ∞ is not irrelevant or immaterial, because it is definitely mentioned in usable reduction rules that can successfully extend classical arithmetic. For example: a + ∞ = ∞ and also a * ∞ = ∞, with for example, a ∈ ℝ. The symbol clearly has some kind of "absorbing" effect on other elements in its domainalcontali

    Good point. It's even in floating point systems. I like the idea of an old cash register modified to ring up

    I also like f() for the limit of f as x goes to . I find this kind of limit fairly intuitive. Many uses of infinity are intuitive in math. Finitism is cute, but it doesn't do justice to our intuition as a whole.

    BTW, you mention math as pure reason. I relate to that formalist point of view, but I suggest that metaphor and intuition are important in doing math. I'd say that it's a kind of language.
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