Aristotle makes this assertion: "For as it is, they have no need of the infinite (for they do not use it), but they need only that something finite can be as great as they want" (III.7.207b.30). Let's first look at Zeno's paradox with the walking to the door example. (It's important to use the word to here because with the word through you would simply walk through the doorway, hence making the paradox absurd and useless). Zeno's paradox seems troublesome to Aristotle because it does not depict the way in which reality actually operates; however, the concept that Zeno was describing is actually a simple, common occurrence in mathematical physics, (especially as described by calculus). What Zeno is describing is called an infinite series. Each successive move towards the door actually represents a distinct function that is mapped out by the infinite series. The series winds up looking something like this:
S1: 1/2....................................................................0.5
S2: (1/2)+(1/4)=(3/4).............................................0.75
S3: (1/2)+(1/4)+(1/8)=(7/8)...................................0.875
S4: (1/2)+(1/4)+(1/8)+(1/16)=(15/16)...................0.9375
And so on and so forth continuing on until infinity. As long as you keep on going another half of the distance, you will find that the total distance traveled gets closer and closer to the whole (1). If one were to use only "something finite that can be as great as one wants," they would certainly have some sort of silly answer, in this case an example would be 1/128. It is only through adding up every step of the infinite series that we see that the true value can be asserted to be 1, despite the fact that the numbers will never truly show it to be as such if you look at any one individual answer.
I strove with this example only to show the importance of the concept of infinity qua infinity. I made the example as simple as possible while still showing the nature of what is actually going on. Hopefully this explanation of Zeno's Paradox will either help your understanding of the nature of infinite series, or raise some interest as to the nature of Aristotle's "potential" infinity in relation/contrast to a mathematically relevant infinity.
Thanks!
ReplyDeleteI'm not a student in Physics, but I wanted to throw this out there. I was just thinking about the interaction between space and time, and Zeno's Paradox only critiques one of these two essential... erm... things. If I'm traveling toward the door at 10m/s (I'm quick!), then I'll be 5m in .5s, and 2.5m in .25s, and 1.25m in .125s, and onward to infinity. But, given that 1 second does indeed pass, it seems that I will indeed reach the door. So mathematics can only be used to disprove the existence of motion insofar as it rejects time. It seems also, then, that if one can claim that an actual infinite exists mathematically, we still have to subject it to this same weakness. The actual infinite exists artificially, as a product of a theoretical suspension of one and sometimes both critical components of reality--space and time.
ReplyDeleteZeno's paradox in no way disproves motion; he failed at that. However, if you are traveling TO the door, (I'm using capital because it won't let me italicize), you would not travel at 10 m/s the whole way to the door. To stop yourself AT the location of the door, you would have to slow down as you traveled. Thus, it would take you an infinite amount of time to TRULY occupy the same space as the door, (especially if we imagine the door as closed, seeing as how two objects could never occupy the same space).
ReplyDeleteI will submit a comment on the concept of infinity in space in a separate comment because it requires more complex math and will look better on its own.
The object I will be talking about throughout this comment is called Gabriel's Horn and will henceforth be referred to as GH. It would probably be helpful also to look up GH on google images so you can picture the way it "expands" towards infinity.
ReplyDeleteThere is an object in mathematics that displays an infinity of "space" in a very interesting way that I think will pose some complex questions about the way in which Aristotle's infinity can be applied and work with mathematical infinity. GH is an object with infinite Surface Area but finite volume. Using calculus functions (integrals), it is possible to determine the volume and area of any object. In the most simple sense, integrals are often the area beneath a curve, (this is how the concept of the standard normal curve is approached in statistics). By involving the integral with formulas for surface area and volume, you can use the formulas to determine the area of something's surface of the volume contained within its boundaries. GH extends towards infinity. The nature of the shape is 'convergent' towards "itself," so the volume has a 'limit' approaching a particular value. However, since the boundaries of the shape are extending into infinity, the surface area of GH is 'divergent,' and it also extends towards infinity. If you were to simply use a big number in this case (as Aristotle suggests), you would have an object with a finite surface area and a volume LESS than its true value.
What the volume of Gabriel's Horn?
ReplyDeleteTucker and I discussed this at length today, and I can't say that I completely understand all of the consequences that discarding infinity might have in nature. However, correct me if I'm wrong, but it seems like there may not be a discrepancy. It seems like Aristotle only discards infinity as it exists among natural things. He says, and I think rightly so, that there is no infinite body, or infinitely divisible body. He does still grant it theoretical, and perhaps, mathematical existence. In that sense, it could still be used in a way that is beneficial in nature without actually existing in nature.
ReplyDeleteI agree with your interpretation of what Aristotle is doing with infinity. It seems that if infinity were involved with natural things, nature would not be intelligible to human beings, meaning we could not be students of nature.
DeleteThe consequence of this is that mathematics probably doesn't help us with physics/nature, but I agree that Aristotle is not completely discarding mathematics' use in other arenas.
Perhaps this is unsophisticated of me, but I just don't see this consequence as being unnerving. Am I missing something?
Trey: The volume of Gabriel's Horn can be anything in the same way that a cube can have any volume according to whatever its dimensions happen to be. The important fact is that GH has a finite volume, whatever that volume may be.
ReplyDeleteMatt: I think I would agree with you, for the most part at least. Aristotle makes a compelling case for there not being an infinite body, and Trey's point about space suggests a good reason to assume that nothing in nature is not significantly infinitely divisible. (For instance, there are an infinite amount of points on Earth, but since there is no thing that could possibly occupy simply a point, seeing as how that would require that thing to be dimensionless and surely no thing could possibly be dimensionless, it follows that the occupy-able space on Earth is not infinite.) My quarrel with Aristotle is his condemnation of infinity's usefulness in math. The assumption of infinity in mathematics allows us to create a more accurate model of the way the world works, and this remains true whether an absolute infinity of being is possible or not, for we would lose our ability to observe and explain much of the finite.
This being said, I find myself wondering if Aristotle would have made the remark I cited in my first comment, (albeit Aristotle did make the remark almost in passing), if he had a working knowledge of modern math.
Where did Aristotle deny infinity's usefulness in mathematics? And that's an honest question, because I could imagine such a claim on his part, but I don't recall him being so explicit. I read the extent of his claims to be denying any possible or actual existence of an actual infinite. Mathematics, though it is useful and succeeds in modeling reality to a stunning degree of accuracy, does not EXIST (substitute for italics)
DeleteHey all, I read through your responses and I'm fairly baffled as to what to add. I'm not sure if Tucker's question is fundamentally something like.. "technical mathematics points to the real existence of infinity" or something like that but...
ReplyDeleteI thought Trey made an interesting point. "So mathematics can only be used to disprove the existence of motion insofar as it rejects time." -- Which to me shows that although on a calculator, the distance between two points is infinitely divisible into infinity, this type of calculation takes place almost in a vacuum, outside of time. --- In Aristotle's terminology perhaps.. the 'infinite series' or 'Zeno's paradox' simply plays on the mathematical imagination as we said in class. In the realm of thought (or mathematics)- the paradox seems irrefutable but, as Trey said, in the actual world of experience, there is a moment which the space between A and B is breached.
Although I don't quite understand it.. something that stuck with me from last class is that Aristotle wants to eliminate the notion of infinity that leaves us with paradox's and contradictions. Further, by denying the infinite, 'you are allowed for things to be determinate' -- suggesting that nature is not a whole or complete?
I agree, the actual infinite may be mathematically useful. It certainly is used today.
ReplyDeleteIt is worth noting that it wouldn't have been useful to an ancient greek mathematician, who, as A. says, only needs numbers to be potentially bigger or smaller depending. In fact, irrational numbers like the square root of two were not considered numbers. Pythagoreans suppressed the fact that there were irrational numbers. Why? It is worth examining. I offer some thoughts below, though.
The square root of two becomes useless as a "number" and only makes sense geometrically, i.e. as the diagonal of a square with integer sides. This means that abstract numbers give way to proportions, ratios, geometrical relationships. Ever since Descartes invented the coordinate system and so algebraisized geometry, the reverse has been true. We reckon geometry in terms of abstract formulae, not concrete visible relationships. In fact, Descartes coordinate plane may be an origin for our current (false) idea of space.
The best book on this is Jacob Klein's "Greek Mathematical Thought and the Origin of Algebra"
ReplyDeleteI think it's really interesting what Bronson said about "denying the infinite". Would Aristotle think that humans are more inclined to accept or deny the infinite?
ReplyDelete