Considering our discussion in class today, I've found myself mulling over a few different thoughts, all of which come from a primary consideration: the difference between Aristotle's conception of nature and a mathematical (contemporary) one. For example, a set of coordinates describes the location of an object based on a grid that we imagine spans the earth. New York City is roughly 40°N by 74°W, but there is not some ontological reality that is this coordinate set. The reality is New York City, the coordinates describe it as it relates to other things on earth. Another example is our concept of velocity: I might move from my chair to the door and think "hmm, I didn't move very quickly." But if I apply a mathematical perspective, I might say "in my movement from my chair to the door I sustained an average velocity of .3 meters per second." On the surface, it looks like all that math is doing is helping us to be more accurate in describing the world.
Here's my primary inquiry. Obviously the tool called math can't apply to things like "that-for-the-sake-of-which," ethical claims, and so forth. But it seems that concerning those things to which it can apply, it necessarily applies with greater accuracy than any alternative system.
Might we assert that any time we are able to make a claim about something using a mathematical system (in locomotion, for example), a similar claim that does not utilize this system is always inferior?
To me I believe Aristotle would say just because we view math as a means for a more accurate description, we only have this idea of it being more accurate because humans "discover" or "invented" the concept of mathematics and measurement. The reason we can say "average velocity of .3 meters" is because mankind invented these numbers to measure distance and time (which also both existed in our imaginations as a means to describe motion). In our minds the mathematical system is accurate in describing motions but that is only because math exists from our own theory of it. In the physical word ".3 m/s" as a measurement of velocity doesn't actually exist and therefore is pointless to use to try and the describe motion of forms. So when we remove our laws of mathematics we have created the more accurate description or perception of your movenment is "not very quickly".
ReplyDeleteI agree with what you're saying here. I suppose I was just wondering if, granted that we see math as a toolkit, there is any way of better expressing a movement, insofar as it CAN be expressed mathematically.
DeleteI'm beginning to see more clearly how Aristotle so different than physicists of our day. It seems like physicists express everything in terms of formulas, whereas Aristotle is widening the scope of things we are able to express and still call it "physics." For example, that he even mentions (in our most recent reading) the following: "They think that void is a cause of motion as that in which something is moved" (214a26), and then proceeds to deduce that "void could not be responsible for the change of place," (214b18) and therefore imply that since our concept of void is responsible for nothing, it cannot 'exist.'
He is going beyond simply quantifying things; rather, he has denied the existence of 'things' that are not responsible for anything. So we might say, as an Aristotelian physicist, "there exists in horses a desire for carrots," but we cannot say that as a contemporary physicist without jotting down an equation about some chemical function in its brain, when the equations don't really exist at all.
Yet, my original question was an advocate for this latter sort of physics, because it seems that where math CAN be applied, it is always so much more effective than any other "tools" we might use.
Oops, excuse the typos.
DeleteSide note: a contemporary physicist wouldn't be calculating a horses desire for a carrot, that's a different discipline. It still uses math, but the math of neurobiology, specifically as it relates to animals. I want to point that out because it's part of what shows the leap in knowledge since Aristotle's time. In Aristotle's time, if I understood Davis' comment on the last post, mathematicians only needed rational numbers; after all, they were only doing geometry, logic, and algebra. We live in a world of knowledge that Aristotle could probably not even have dreamed of.
ReplyDeleteThat we did invent math is part of what makes it so incredible as a modeling system. It's a universal modeling system that can be applied, tested, and advanced by anyone in the world. If someone in Japan were to discover an intrinsic flaw within Newton's Law of Universal Gravitation, it would be fixed and applied to all of the scientific community within a few weeks. What's amazing about math and physics is that they are both integral to our ability to understand the physical world. Without that concept of velocity, especially in relation to the forces of gravity and friction, we would never have airplanes. If we were just to analyze things using terms such as "really fast" and "really heavy" and "lots of exposed surface," we would be forced to approach the world only through trial and error. Math is no more "invented" than language. We would have no way of describing the motion of something if it weren't for language. Math is just an alternative way to model something. Both math and language were created so that we can model the world, and if language had been enough on its own, math never would have been necessary to have been invented. If we want to be technical about existence, as long as math is excluded from existence, you must also exclude language. It'd be really interesting to try and describe nature without quantifying the description with words or numbers, (this is where the importance of art would come in, but even art is a human construct of a sort).
I have to agree with Trey that in things that we are certain that our formulas model correctly, (such as gravity, friction, energy conservation, etc.), there is no model that gives a solution as exact or applicable. Math cannot describe all things, (yet, just because we don't understand the math behind something doesn't mean it might not be there), but insofar as it can describe things, it is more useful than anything else in our "toolkit."
Maybe the reason we are so elitist towards language as opposed to math is because we don't all know math at a level capable of modeling the world. It's the job of the scientists using math to figure out the world on their end then pass the knowledge down to us through language. If someone straight up started describing things mathematically, 99.9% of the population would get very lost very quickly (including me honestly). We use mathematics as a route to discovery then are able to teach the things we learn from math to the world without them ever necessarily knowing that math is where their understanding is coming from; however, we would not have been able to attain many of the levels of understanding we currently have if it were not for the math that got us there.
I apologize for the length of that comment. This little box makes it seem like a lot less when I'm typing. I'll do my best not to go rogue like that again.
ReplyDeleteHa, good descriptor.
DeleteI don't think we can say that any claim that could be made mathematically, but is not, is always inferior.
ReplyDeleteWhat if we're trying to describe the complex emotions your movement from the chair to door made us feel? We could describe it mathematically, as 43% graceful or beautiful, but that doesn't accurately capture the depth of experience.
In the last chapter of Book III, Aristotle dismisses simply basing things off of thought. He says that anyone could simply think that they are bigger and bigger, but obviously they would, in reality, never be bigger than the city or than they actually are. Just because you can assign an arbitrary value of percentage of "43%" to "graceful or beautiful" doesn't mean that you are actually mathematically making your claim. To my knowledge, we do not currently have a way to mathematically map it out; therefore, like Trey said, math would obviously not be the correct "toolkit" to use to understand something's gracefulness or beautiful-ness.
ReplyDeleteI agree with Luke's comment about the depth of experience. In my opinion, mathematics can only take you so far. How can mathematics ever be completely accurate when it doesn't account for the way something can affect us emotionally? I think the depth of experience that occurs is essential and without it an accurate description is impossible.
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ReplyDeleteWell as far "grace or beautiful" is concerned, mathematics (in some quasi-form) has an obvious entry point in the study of neuroscience. Clearly a statement like, "43% graceful or beautiful" not only sounds ludicrous, but also seems rather unscientific - so here we draw the limits of Math in unison.
ReplyDeleteLets say however a series of studies conducted came to a conclusion that feelings we give the name "grace" or "beauty" are often associated with [x number] of milligrams of [given chemical compound in brain]. Who knows, maybe there's even a proportional relationship at work. Suddenly it seems, mathematics has a way into intimate corners of our lives.
A lively debate.
ReplyDeleteAristotle's comment about "thought" at the end of book III that Tucker mentions is worthy of attention here. One problem with mathematics as a tool for the study of nature is that it allows us to get carried away in our imaginations, our thinking, without a constant return to observation.
The power of mathematics comes from the way it can assess all units and relations on the same ground (by reducing them to a common value, say, 1). 1 apple = 1 orange mathematically, as units. But 1 apple does not = 1 orange in nature. In short the proliferation of qualitative difference in the natural world makes it unmathematical. Words may also "reduce" the complexity of nature. nevertheless, language, aided by reason, may come closer to showcasing and evaluating the diversity and the varieties of unity observable in nature.
Notice that mathematicians and platonic philosophers both exclude Material (book II, ch. 2), the principle of difference, in order to understand nature on a univocal foundation. Aristotle aims to include it.
Wells: That very thing has actually been done with depression. Studies have connected it to an absence of a particular chemical, and now patients are better able to be assisted by doctors as they "come out of the darkness."
ReplyDeleteDr. Davis: We can get just as carried away in language as we can in mathematics. 1 apple would never be compared to 1 orange qualitatively, only through quantity. The rules of making sure you use the correct units, and that those units are proper, are an integral part of making sure that the concepts of physics are correct. A simple unit mistake such as using grams instead of kilograms (both of which measure mass) can ruin all of the output of a problem.
I think the primary concern with the quality vs. quantity distinction is manifested most poignantly in Aristotle's rejection of void. I agree with you that if we simply keep a close eye on how we are describing things, we will not confuse 1 unit with 1 type of thing. But what if the entity disappears? Then we neither have 1 unit, nor 1 type of thing... but Aristotle's contention is that we ALWAYS have some type of thing. The seduction of describing things mathematically makes us think that if we don't have 1 SPECIFIC type of thing, we don't even have 1 unit. So we conflate... (once again, caps in place of italics)
DeleteWhat's so great about keeping a QUANTITATIVE void for the sake of mathematics is that it allows for a modeling application of Aristotle's concept of potentiality. Without a quantitative void there would be no way to account for any THING full of potentiality in mathematics. The more I think about it, the more I think there is actually 0 (pun totally intended) difference between being full of potentiality and the quantitative void I am asking for here. We would still call a qualitative void ridiculous, especially now with the examples of becoming that Aristotle has discussed in Book VI.
DeleteIf I am understanding Aristotle correctly, place must always be bounded and talked about in relation to motion (212a5 & 212b25). And motion is determined by the material of a form. So, while I understand the value of speaking mathematically sometimes, I do not think we can talk about Aristotle's conception of motion mathematically because, like Dr.Davis notes, the language would be derived from our imaginations instead of from our actual observations of nature.
ReplyDeleteBut doesn't all of our "natural" language also come from the same faculty as does our mathematical language? It seems the real danger is exemplified in the fact that we 'know' what 0 is, mathematically. It is 0, plain and simple. But imagine encountering 0 in the universe... that's unthinkable. Aristotle seems to attack mathematicians because many of them feel that 0 is as existent as an apple (to continue the metaphor). But that leads to all sorts of wacky ontological misconceptions (Zeno's paradoxes, for starters).
DeleteI agree with the first sentence of Trey's comment here. I also think that we have found (which I already said in a different way in my reply to Trey above) in our discussion of Book VI that 0 is in fact full of potentiality. As long as we remember to think of 0 as "begging" for quantity, or activation of its potential to hold value, we will be avoiding all of the misconceptions we're currently so scared of.
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