At the end of class on Tuesday, Dr.
Crist suggested that I volunteer for these readings due to the philosophic
implications discussed, probably hoping that, being a philosophy major, I would
be well equipped to find and display said implications for discussion. To be honest, in one article I was frustrated
by my limited knowledge of Set Theory and in the other I couldn’t help but
recoil at the seeming dismissal of philosophy by many of the thinkers discussed. Most notable among these men was Veronese,
who is portrayed with the opinion that “If, [Veronese] suggested, philosophy is
intended to be the highest level of research into truth, then mathematics does
so much better than philosophy ‘being not only the most ideal but also the most
positive of the sciences, because it is the oldest and the most precise
expression of the truth.’” Ouch. Certainly our kind author should offer some
sort of solace for the wounded philosopher: “But is every cardinal of the form
of a natural number? … More food for the philosophers.” As if they’re animals at a petting zoo.
Anyway, putting that behind me,
there are some serious philosophic implications to each of the readings that
deserve discussion, the first being the comparison of shapes in mathematics to
the idea of Plato’s forms. Plato’s forms
are ideal beings (in that they exist somewhere
in reality) that act as a standard for other beings in the real world to “participate
in,” functioning much like a concept. An
example is the idea of Beauty. A thing
can be beautiful, but it is never as beautiful as Beauty, and it is only
beautiful because it “participates in” Beauty.
Other examples of the forms are Goodness, Hotness, and pretty much
anything else you could imagine.
Contrast these forms with mathematical shapes and numbers. Perfect squares and circles don’t exist in
our world, but it seems plausible that they exist within reality. This notion, considering a conceptualization of something as a validation for its existence, can be traced all the way back to
Parmenides and his ontology. So I guess the
following questions arise: 1. Do the forms exist? 2. Do shapes and numbers
exist? 3. If we assume the existence of one, must we assume the existence of
the other?
According to Hermann Weyl, math is
not true because it is logically consistent, but it is true because it exists
within Physics. That is to say, we can
use math to express and interpret the world around us. Suppose mathematics did not accurately
describe the real world in tandem with Physics, but was still just as logically
consistent as it is today. Imagine if
Newton never connected Calculus and Physics, but with a disjunction infinitely
wider going all the way back to Euclid’s geometry. Would math still be true? Or do you need a grounding in physical
reality for truth?
I’m not sure if because it’s midterms week, or the length of these readings but I caught myself rereading most of these two articles. What really stuck out and made me think was the aspect of measuring time and the difference in measuring length by Poincare. How we created and began to follow time by allowing our senses and our unconsciousness. In response to your questions, I literally put my hands up almost surrendering this whole post. That’s when my roommate stepped in and we began discussing these questions while having dinner. We discussed how numbers exists to this point into where our consciousness forms this perfect image. Discussing also how our brains and minds will unconsciously makes these forms seem perfect. She was stating how it is because of how we are taught; I more believe it’s a combination of the need for us as humans needing to feel this sense of complete as humans. As for if we assume one exists she started bring into account physics and how it’s possible to show the 2+2 = 5? Completely blew my mind, currently losing all hope in what I thought I knew and what I will be teaching my future students. This was a great way to end midterms week.
ReplyDeleteAbbie, I'm glad to know we're messing with your head.
ReplyDeleteI especially found the second paragraph you wrote about the forms that other beings "participate in," to be very interesting. I had never thought of Beauty being a form, or that it is a form that could be thought of in a similar context to mathematical forms. In the reading on measurement, the author discusses measuring the double of something. He uses the example of pleasure being something you can have two of but both together do not create a new pleasure. These intangible things being measured and put into similar contexts as mathematics puts the idea of mathematics and measurements into a whole new perspective for me. Measurements of length and such had always seemed such a concrete concept to me. I think this also coincides with the idea the author presents that science is constrained. I think it is amazing that mathematics went past this and became a means to challenge these constraints to human thought.
ReplyDeleteZach, I enjoyed your discussion regarding whether or not the forms of beauty and goodness could actually exist in reality. Mathematics is oftentimes considered to be “the oldest and most precise expression of truth” (340). What is “known” in other sciences is constantly being refuted and revised. Once something has been established as a mathematical truth, however, it can never be refuted. This corresponds to the intuitively rigorous nature of mathematics we discussed last class. Along these same lines, Gray suggests that mathematics exists as an entity superior to philosophy. However, I do not necessarily agree with him in the sense that not everything in the natural world can be modeled and quantified numerically. You bring up a good point in your discussion of the forms of beauty and goodness. It’s interesting to question whether or not these ideals can actually exist in reality. Gray presents a similar example in his discussion of happiness. He suggests that although I may be able to quantify my happiness to the extent that I can be much happier today than I was yesterday, I would get nowhere trying to assign numerical significance to these feelings. One of the things that seems to particularly frustrate mathematicians is the idea that math is almost “too perfect” of a science, and thus it can never truly model the natural world. In the natural world, things simply do not behave as perfectly as they may be projected to behave in the mathematical world. A model may be mathematically perfect, but when it comes to real life experimentation, actual values tend not to match this projection perfectly. Just as we cannot necessarily quantify or identify the form of beauty or goodness, our mathematical approximations of the natural world may never exactly reflect natural realities. Paradoxically, it is because of the idea that works too perfectly that it can never be a perfect representation of what we know and experience. It is important to consider mathematics in conjugation with philosophy and the other sciences then, in in order to truly comprehend our life experiences.
ReplyDeleteI thoroughly enjoyed your post,Zach, and was thinking that the readings did seem a little biased towards math and not as much toward philosophy. The concepts of measuring different things and modeling the world from mathematics didn't really confuse me as much to the same extent the other posters mention, but then again maybe having a quantum phys. class with Dr. Duda actually helped a little. On the topic of math being too perfect, I think the case is exactly the opposite. Mathematics might be the most sound and logical of all of the sciences, since it has to go through such rigor to be valid, but math itself is still growing and changing. Many ideas that math has birthed have been applied to the physical world and some physical phenomena have yet to be described by math. I see this as a flaw somewhere between our understanding of the world and our "idealized" form of mathematics. There is so much more to the world that people haven't made connections to yet. As I believe Nathan said in class one day, it doesn't seem too far fetched that we are all connected by the same cosmic fabric. So why then haven't we been able to connect all of the sciences together with math? And this is what I see as a flaw.
ReplyDeleteI agree with Jamison in how Zach's post was really fun to read, but I am also with Abby as where my brain just wasn't working when I read these articles. Although, I did enjoy the part about Hilbert and the introduction of the + and = sign into mathematics. Just because I never knew any of that information. When we were kids we we always taught 1+1=2 or 1+2=3. We were never explained why or who invented any of this. Along with upper level math courses I have taken, we always just assumed that was always correct and never really found the need to know who invented that. I guess this just interested me because it related our talk we had in the previous class, where if math was explained more and made more relevant to my life (everyday life) then maybe I would have been more interested and been a math major.
ReplyDelete