By cutting a line the only thing you can get is line segment. As much you cut, you never get to anything like a point. But, how do you cut a line if not at specific point? If there are no points, then there can be no line segments either.
By cutting a plane, the only thing you can get is plane segments. As much you cut, you never get to anything like a line (or curve). But how do you cut a plane, if there is no lines? If there are no lines, there are no plane segments.
Same goes for space.
A brood comb 
I would think you cut a line between points, not at a specific point.
Maybe, but I’m still confused about this :). I used to think that there is no such things as point at lines, but only segments, but now I can’t even see how can there be segments, if there are no points.
Interestingly this is key to Peirce’s philosophy. (Yeah, an odd thing to be key, but it’s how his notion of continuity differs from say Cantor’s infinities) It’s a chapter in his Logic and the Reasoning of the Things with two great introductory essays explaining a lot of it.
By ‘cutting’ a line sure you can get to a point. A line is an abstraction in just the same way as a point is..just as a plane is. This distinction between conceptualizing space and actual space is also the key to Bergson’s philosophy. Space itself is indivisible — lines and points and segments are not even applicable to real lived space. The idea of line, point, plane, these are issues for mathematical geometry.
Not necessarily. It depends a lot upon how you do it. There are different mathematical practices each with rather different results.
If you think of open segments like ]0;1[ you don’t need any points that exist but only points as an abstraction.
There is no need for a point between the two cut peaces ]0;1[ and ]1;3[.
I do not know the solution of Peirce or Bergson. Is it something like this?
Thanks Clark, I think I will try to find that one. Sounds interesting.
Trey, I’m not sure I buy into such separation. If those are abstractions, why wouldn’t we be able to see them as abstractions from real space (not that they exist as such in real space, but that we can consistently think of them as aspects of the real space)
Fangtunsdoch, interesting, wouldn’t be something missing from the line if we cut it to two open segments? Or maybe this question is just bad intuition on my part?
yeah…what I mean by an abstraction in this case is just that it is pure mathematics (as opposed to ‘applied’ mathematics). Is it a point..is it a line…it’s relative to your frame of reference. And like you aptly mention..there are different ways of cutting a line in mathematics. There are no lines nor points, strictly speaking, belonging to the furniture of the world..if you will. At least that is my conjecture. However, I concede that others may disagree…
No, there is no solution for this problem in Bergson. The reason is because Bergson saw it as the quintessential mistake of most philosophical and scientific thinking that it asserts a metaphysical or ontological correspondence between pure mathematics and actual reality. I know you don’t want a run down of Bergson on here…but the idea goes all the way back to Zeno and run right up through Newton and Descartes right…so how does Achilles run the race? Of course we know..if we apply the same standard to the race track as we do to a line then not only can he not finish the race..but he can not even get started.
My general tendency is that mathematics is not ‘innate’..either to our conceptualizing a world nor to the world itself. Although I would tend to agree that certain logical relations, although also not innate, are extremely probable to develop in terms of how we put together a picture of the world from our point of view (‘if..then’ ‘and’ ‘not’)…mathematics is presumably a product…at least in its origins..of these logical frameworks from which we make sense of the world. So..1 + 1 = 2 and cannot even possibly equal anything else…why? Because of the very framework from which we make sense of our reality. Can you just say I chose not to understand reality from that framework? Absolutely not…this framework began to develop the minute you began to have experience and your brain developed along with your experience…and so forth. Does this mean that this is reality as it is ‘in itself’..absolutely not. But it does mean we have very strong ties to this way of thinking about the world…thus the only people who really have any problem with it are philosophers…and some artists I suppose.
I’m not sure if that got at the question you were asking…
I would agree with Trey that points, lines and mathematical surfaces are only abstractions. (In order to simplify matters think of the current three dimensional world like a one dimensional world: so points are the only candidates for abstract objects.) But I also like your question, Tanasije, whether there is missing something. Some ideas to cope with this question:
a) When cutting you will always have a cutter with some extension. So before cutting there will be only a interval like ]0;3[ and after the cutting two intervals ]0;1[ and ]1.1; 3.1[ and in between the blade of the cutter ]1; 1.1[
b) When you say that the real world is at most countable infinite, but its models are uncountable, there will be no difference in the real word between something concrete corresponding to an open interval (like ]1;3[) and something concrete corresponding to a closed interval (like [1;3] ).
c) As far as I know there is also an isomorphism between the set of all open intervals and the set of all closed intervals. And there is a isomorphism between the set of all open intervals and the set of all half-open intervals. Because there is no difference in measuring, you may cut also in a way, that the cutting point is included in one interval, p.e. ]0;1[ and [1;3[
d) Intuitive idea: the measuring is o.k., that’s the purpose of mathematics, that’s the purpose of the introduction of points. Points have no further use, so forget the idea that there is a real point when you cut a line (unless you do think of the lengths of the cut peaces or unless you want to use differential calculus).
You can get to points by cutting a line if you do a supertask, which is easy to do in pure geometry if not in the real world. You cut a line [0, 1] at 1/2, then at 3/4, 7/8 and so on. You end up with line segments [0, 1/2], [1/2, 3/4], [3/4, 7/8], … and one point at 1 left over. Similarly if you use open or half-open segments.
Another supertask gives you points everywhere. You cut at 1/2, then each of those two in half, then those four in half, then those eight and so on. You end up, after the supertask, with no unbisected segments. Benardette discusses such things in his “Infinity: An essay in metaphysics.”
Points are weird. It seems they must exist everywhere in continua, if they can exist anywhere; and it seems that a point can exist (conceptually or logically) since there is a point at the corner of a square. And yet if they are everywhere then they are everywhere in the line next to a point, that goes all the way up to that point; and yet none of those points are next to it.
Weird. My provisional position is that although points are everywhere, lines must be closed; that an open interval is just the whole closed line with additional properties at the end-points. But I’m unhappy about that position.
Hi EnigMan,
It is not so easy to do your supertasks to get points. You have to do infinite cuttings. In mathematics you use the limes function (which is e.g. always used in the differential calculus). I doubt that something like this can be done in the real world.
I would prefer that the world is a finite world with minimum units in time and space. But I also think as long as the world is enumerable I will have no problems with cutting. As far as I see you think that the real world is innumerable and that there are things that correspond to the real numbers: in this case you accept the existence of points in a nearly trivial manner.
Enigman, that seems a very interesting point to me, when you say there is a point at the corner of the square. Maybe we can say that the points can exist conceptually or logically only due to the conceptual or logical assumption of lines themselves (where we imagine them not having any width, and when we get two lines crossing each other – we get a point). But then, are lines conceptually possible? In same way, further, if we think of the lines as what we get at intersection of two surfaces, maybe we find lines conceptually and logically possible only due to the assumption of conceptual possibility of surfaces. Not exactly sure what to make of this… :-/
Fangtunsdoch, I like that you are relating the issue to that of measurement in “real space”. I kind of believe that what is unreasonable (or has contradictions) logically and conceptually can’t be actual too, so I think if we find that something is unreasonable in our ‘abstractions’ like points, even less we should expect it to find it as such in the actual state of the world. And I wonder if maybe when physics wants to speak about determinate coordinates, it doesn’t actually bring in unreasonable abstractions into the real world. So to say, maybe quantum indeterminacy shouldn’t puzzle us much. Maybe there is no sense in which one could talk about determinate positions. Again, I’m throwing just vague ideas here…
Trey, I tend to agree that we build such kinds of framework, but I think that any framework will require some basic notions – “somethings” that we can think about. And that we can’t think of the world except in those terms, so that some kind of Kantian “skepticism about knowledge of the real world” I think is conceptually impossible.
I don’t think we build these frameworks..I think these frameworks develop in conjunction with our relationship to our environments from day one..and before (and interestingly…an environment itself emerges from this development…this is a core philosophical problem for me..but that aside). Although ‘thinking’ in the ordinary sense is not the fundamental relationship that we have to our environment (then or now)..I am totally in agreement with you that we do require some notions..they are the logical notions (and, if then, either or, not) that structure our understanding of ourselves..our world..so I do take a Kantian approach on that end of things (still with some qualifications..remember I don’t believe our primary relation with the world is representational). Like I said also, I don’t take these notions to be necessary, neither mentally nor metaphysically. As I see it, ourselves and the world could get along just fine without them..just as the world does and most other animals do) but ‘thinking’ could not get along without them..at least not thinking in the sense that I see us doing it. As I re-read over your reply..I think I see a possible ambiguity in my idea of ‘thinking’, so this also probably lies at the core of your questions about my idea of abstraction. I take thinking to be primarily conceptual..but I think we have many, and maybe even most, of our experiences of the world are not primarily conceptual. Mathematics, just like philosophy, and religion, and technology, developed within this…well…not really sure what to call it. Perhaps something like ‘lived world’ or ‘living world’…but those terms carry a lot of philosophical baggage. I don’t like ‘experience’..I don’t like ‘context’…I really haven’t found a word yet that I think really gets at what is being got at here…
Perhaps that clears the path a little for seeing where mathematics lies within my scheme of things…or it either confirms for you that I am a confused philosophical mess!!
I’m somewhat puzzled when the conceptual thought is negated to animals, because I can’t quite grasp what would it that which the animals would lack, and we on other side have *except* that we are aware of lot of more things, and we have a know-how of lot of more things.
It would seem to me, that animals have to be aware at least of things, events (changes), qualities of things, of categories of things (e.g. wouldn’t we allow that they have knowledge of categories of things, that this kind of food is tasty, and that one is not?), and so on.
They would, I think, lack capacity to imagine counterfactual situations, or develop complex abstract arguments by generalizing, but I’m not sure this means that they lack awareness of what you call logical concepts. Why can’t they be aware that X *and* Y? Of course, they will lack the awareness of language, and awareness of the possibility to inform somebody else of those things that they are aware of, but why would we take the ability of such expression to be qualitatively different in such a way that we distinguish it (vs. subsum it under) from the animals awareness.
To be aware of something and to have a concept of something are not the same. Even to be aware of something and to be aware of something and be self-conscious, are very different modes of awareness. To have a concept, I take it, is to have a notion of something that is universal from a particular (or a universal alone..such as is the case with mathematics). They surely lack the capacity for comprehension of logical concepts…although they very well may act in such a way that we (given our frame of reference) apply our logical concepts to interpreting their actions. I see the difference in our way of relating to the world as not only quantitatively different but qualitatively different from other animals. I don’t really see it as we know how to do more things..every creature pretty much knows what it needs to do relative to its unique circumstances. Ultimately the difference lies in our particular temporal relationship to the world and ourselves. So we might ask — why do we have concepts and other animals do not? An almost equivalent question is why do we have a past and a future and animals do not. We are dealing here with an immanent transcendence. Its not that I have a concept and then can transcend my present moment (think about what I am going to eat for lunch today) but that there is this deep inner drive towards..push towards..getting outside of my moment..that allow my concepts vaguely and then clearly to take hold of the past and future.
A square is clearly conceptually possible, Tanasije, I’d’ve thought. One thinks of a square when one sees something that is approximately square, such as a sketch of a diskette. And so lines and corners are conceptually possible.
And if points are parts of continua, then they exist whenever continua do (e.g. time maybe). They are not parts of actual continua like molecules are parts of actual materials, but are more like hours being parts of days.
So I would go the other way round, and say that surfaces are parts of continuous spaces because points are parts of lines.Two lines can cross in a point because points are in both lines, rather than the point existing because the two lines cross.
Trey, I’m mostly skeptical of ‘concepts’ as some kind of mental constructions, so maybe that’s why I’m inclined to draw relations between awareness of something and a concept of something.I will think about this… maybe write a post on my thoughts about that…
Enigman, I’d’ve thought that too. But now I’m not so sure… if I don’t imagine lines as possible, and then imagine one line crossing another line (as roughly happens in the case of square), or some similar situation, can I make sense of a point? The idea is that points make sense (or are conceptually possible) only if we assume lines, lines are possible only if we assume surfaces. Anyway, it is just pretty vague idea…
I think you’re right, that points makes sense only as parts of a higher-dimensional space. If we begin with planes, we can ‘see’ that lines and points are possible, by thinking of a black square on a white background. And surely planes are possible, conceptually and logically. There might be more doubt about whether space is possible, no? The plane only has to exist sufficiently for us to have shapes, so it does not have to exist externally, independently of us, objectively.
Generally, I agree with you. For the most part, concepts are concepts of some thing. My idea is that mathematics is more akin to language…which I don’t think of as fundamentally conceptual. I think that is how we tend to think about the nature of language but I don’t think that is really what language is. I guess it works like this. We develop certain foundational frameworks through experience and through which our experience is interpreted…from such frameworks mathematics develops (along with abstract reasoning in general). So, while mathematics is not about the world strictly speaking…it’s also not just a mental construction either.
and again..that ties right in which how mathematics can be so pragmatically successful in dealing with the world…we are not immediately dealing with the world when doing mathematics…we are dealing the frameworks which manifested from our relations with the world.