Discuss:Mathematics, Logic, Philosophy:Mathematics as Art
From da Vinci Concept
Mathematics as Art
henrik - Mon Aug 2 1:59:29 2004
I have just returned from the annual European meeting of the ASL (Association of Symbolic Logic) in Turin Italy. After one week of countless tutorials, special sessions and contributed talks, the experience left me with the impression that Logic and Mathematics are just another art form like painting, sculpting or literature.
Applied Mathematics is akin to art that has been commissioned by some patron for a particular purpose. An example would be graphic design work that is sponsored by a marketing agency.
Pure Mathematics is akin to "les beaux arts", i.e. art that has been pursued by an artist simply for the sake of art itself. An example of this would be the impressionist painters who kept painting their controversial art despite that they were not making any money off of their paintings and that they were being ridiculed by critics and society.
At some level, I feel that I have known this all along, but this realization that Mathematics is really just another art form has never been as clear to me as it is now. The more I think about it the more I see correspondences between Mathematics and other artistic forms. For example, reading a great piece of English literature is analogous to reading a great proof in Axiomatic Set Theory.
From a philosophical point of view, I find this realization to be quite interesting since it seems to contradict some classical philosophical worldviews. For example, Plato regarded Mathematics as a source of pure, absolute, eternal and universal truth. Whereas Art was considered as a danger for society and believed that, the philosopher king should banish all artists from the Republic.
--henrik Mon Aug 2 1:59:29 2004
sunny - Mon Aug 2 19:51:07 2004
This is an interesting observation. It seems like it should be quite obvious that Mathematics, pure mathematics, can be recognized simply as art. I had never come to this conclusion, always treating Mathematics as a means to obtaining art through its application, but not mathematics qua art.
However, my perspective on Pure Mathamatics as needing purpose and applicability would not falter in regard to this realization. For the same reasons that Mathematics is useful when it achieves application, so can this be said about art. If one produces art without some implicit or explicit purpose, a purpose will likely be found out by viewers of the artistic work. This purpose perhaps deviating from that which one may have intended had it been identified. Similarly, even though the pure mathematician produces art, and often does not identify a well-defined purpose for the product of their efforts, the consumers of one's mathematical work may find higher purpose and application.
I am still of the mind that to intentionally give one's work meaning through identification of purpose will much more likely produce objects of value. If one does not consider intent, purpose, or meaning, it simply means that the intent is to have no purpose.
It might also be considered that human mind and all of the insight that we build into it through experience may also produce the intent and meaning of our work without our conscious volition. This might be seen as the endpoint of learning an 'art', enabling the production of objects which have value, borne out of the complex cognitive network that we ourselves produced through careful, observation, study, and self-refinement.
--sunny Mon Aug 2 19:51:07 2004
joryea - Sat Aug 7 8:11:43 2004
hmm.. indeed. Pure mathematics would be considered art. It wasn't long ago that I had forgot such beauty in mathematics. The memorization of algebraic formulism is, well, the highest achievement of this is something like what Euler achieved. Although this is simply macho masturbatory workings of proofs and such without any actual rigor whatsoever into the actuality of mathematics, and instead left undefined imaginary and impossible numbers and such, or complex numbers with imagnary and impossible formalism within the algebraic equations; leading directly towards a destruction of actual mathematics.
Art is taken as virtually anything in this degeneration of society and culture as a whole, i.e. worldwide. As Sunny stated, "If one produces art without some implicit or explicit purpose, a purpose will likely be found out by viewers of the artistic work." This is the accepted view, in most cases, of the identification of art, or how people percieve it. Truthfully, and to be quite honest about it, if the work which you are currently involved in has no actual purpose towards a growing negentropic universe, i.e. uplifting the culture, and its interaction with the biosphere, as well as the noosphere, then it is not art. The term art, is greatly vomited upon, and i'm sure that Plato's Socrates, our dear friend of times come and gone, would give a beautifully defined meaning to the term that would easily rebuke what is generally accepted. Yes, the mathematical formalism you speak of is taught today in schools and universities, and is generally accpeted by scientists workwide. However, your speaking of a dark age in truth, and the idea there of.
So, in othert words, these so called mathematicians and scientists have lead us into a collapsing culture and a worldwide economic crisis. What's this say about indiginous art, supposed art, and it's lack of actual purpose? When I say purpose, I mean to say, human understanding, i.e. actual, specific, intentional, furthering of understanding of the universe and human existence within it.
If you would like to redefine, or perhaps challenge this view of art and mathematics, I suggest looking into the work done by Carl F. Gauss, Riemann, and such beginners of calculus as Da Vinci. On the quadrature of the circle, I suggest the king of the ranaissance, Nicolus of Cusa.
[ Edited Sat Aug 14 2004, 11:15PM ]
--joryea Sat Aug 7 8:11:43 2004
Truc_Ha - Thu Aug 12 4:49:20 2004
Joryea:
Your post, though colorful and firm, is incoherent, making it unclear what you are saying. I hope that you will consider the following suggestions in order to facilitate your contribution to the discussion.
Two general recommendations:
As I have been reminded before in my own writing, philosophic discussions are usually very dense, full of sentences that are statements meant to be read. Therefore it is helpful to break up exposition into smaller, more manageable paragraphs. This is analogous to pausing to breathe.
For these paragraphs having a succinct topic sentence somewhere in there, not necessarily the end or beginning as is advocated in school, is also worthwhile. It gives the reader a chance to read what you want to say/argue in the paragraph and decide whether or not you have done so successfully if at all.
More specific recommendations:
The word "this", like the words "its", "them", and other pronouns are pointers that indicate something else. What is very confusing in this post is what exactly you are pointing to. For example, the last "this" could really be pointing to any number of things including (according to your post) the degenerate views of what is art, of what is math, of truth, etc. Be careful with pointers in philosophy; as in programming if they aren't explicit, the overall piece/program breaks.
You use "i.e." A LOT. Yes, I know it's "id est" in Latin, and as a (beginning) student of Latin, I do not object to it, but I suggest mixing it up. Also, in the sentence "What's this say about indiginous [sic] art, i.e. supposed art and it's lack of actual purpose?" I question you implication that "indigenous art" is the same as "supposed art".
I applaud your new tendency to read and mention other works; it is a refreshing change. I recommend embedding links in your posts to, say, Amazon for books, so that everyone else has a chance to look up these names you're dropping and judge for themselves those people's ideas.
Finally, a pet peeve of mine. Please SPELLCHECK your work.
Thank you.
--Truc_Ha Thu Aug 12 4:49:20 2004
joryea - Sat Aug 14 23:31:35 2004
Thank you for your advice Tru-cha, it is quite helpful. I proofread again and tried to correct errors where I saw them, yet I'm not quite sure where you don't understand where the work becomes incomprehensible. If you could point this out more explicitly for me I would greatly appreciate it. I do tend toward larger sentences.
Truc_ha, is there a spellcheck on this website? If there is, how can it be activated? If not, I'll remember to type up any responses and such on a program where spellcheck exists. I can't stand it either truthfully. I think I just get lazy.
About the books on these beautiful minds: I don't know very many on-line sites for this kind of information. Although, I'm sure if you look into any of the mentioned names, you could find out numerous amounts of information. About the biosphere and noosphere, I recommend the teachings and work done by Vladmir Ivonovich Vernadsky. I will look into finding some exact methods of getting this information in reach of easy access.
for now, you can look into http://www.wlym.com although it doesn't have work by Vernadsky, Da Vinci, or Euler. Although some information might be achieved about there works within this website.
http://www.21stcenturysciencetech.com is also a good sight for such research.
Um... a lot of these guys works are not yet translated, and those works translated are hard press to come by sometimes. The schiller institute has been working hard in translation of many peices for a long while now, and so some are available through these above sites, and in other areas of research. http://www.schillerinstitute.org is a good sight for renaissance like material if anybody is interested.
If you have any specific questions about any of thee above information or wish to begin a dialogue, you may contact me at my listed e-mail address. I will try and find more sites for further development, and perhaps some books, as you suggested, at amazom or something. Thank you.
[ Edited Wed Aug 18 2004, 05:41PM ]
--joryea Sat Aug 14 23:31:35 2004
Truc_Ha - Tue Aug 17 18:49:11 2004
| Tru_cha, is there a spellcheck on this website? If there is, how can it be activated? If not, I'll remember to type up any responses and such on a program where spellcheck exists. I can't stand it either truthfully. I think I just get lazy. |
| -- joryea |
At the very least you could spell my name correctly.
--Truc_Ha Tue Aug 17 18:49:11 2004
grefrath - Wed Aug 18 10:21:22 2004
Thanks to everyone for your posts on the subject, it was a pleasure to read them this morning.
Iíve always felt that literature, and poems especially, are a type of incantation, a type of spell. For the root of poems was the spoken breath, not manuscripts. I have thought for quite a while that, similarly, art and mathematics are intrinsically linked. That is, if manipulations of breath can alter mood in the case of reciting a poem, just as huffing quickly in and out can quickly produce agitation, just as breathing before sleep has a rhythm more shallow, then it seems to follow that this effect is not only art, but that this is a form of mathematics. It can be reproduced by reciting a work again. Anyone who knows the lyrics to ëItís the End of the World as we Know It,í by REM, and sings along, quickly understands that a sort of algebra of breath appears, which induces excitement (at least in this writer).
But it was interesting, because in Henrikís statement, he is stating actually the reverse of what I had previously believed. Not that art is mathematics, but that mathematics is art, which is very curious for me. I speak entirely as a layman, but reading Sunny's response, perhaps the beauty in applied or pure mathematics, the art so to speak, is looking at how a problem is solved, the technique. Perhaps a similar parallel (though perhaps it is not, and would be eager to hear response), is to read only the last two lines of a sonnet by Shakespeare, without reading the work in its entirety. One would get the gist, but would miss out on how the artist/mathematician achieved this conclusion in their thinking.
As far as the difference between applied and pure mathematics, it would seem this is the same problem that artists always deal with, how to find an audience (i.e., sell oneís work so that one can support oneself), versus doing experimentation which keeps the soul growing. If one wants to engage in ëpureí mathematics but there is no audience for it that one can find, then one could probably occupy oneís spare time with it, but one would not expect to survive on it necessarily (or at least not survive very well).
Many thanks again for the postings
[ Edited Wed Aug 18 2004, 04:49PM ]
--grefrath Wed Aug 18 10:21:22 2004
joryea - Wed Aug 18 17:47:44 2004
What is mathematics?
--joryea Wed Aug 18 17:47:44 2004
henrik - Thu Aug 19 7:16:05 2004
"perhaps the beauty in applied or pure mathematics, the art so to speak, is looking at how a problem is solved, the technique. Perhaps a similar parallel (though perhaps it is not, and would be eager to hear response), is to read only the last two lines of a sonnet by Shakespeare, without reading the work in its entirety. One would get the gist, but would miss out on how the artist/mathematician achieved this conclusion in their thinking." grefrath
I cannot speak for other mathematicians or other students of mathematics, but for me the beauty experienced by reading and understanding a proof is in seeing how everything fits together. You start out with some premises. From each of these assumptions comes out a cascade of information, which gets combined at exactly the right moment, which generates even more information and eventually leads to the conclusion. And yes, it is frequently the case that it is not possible to understand every single step in the proof, but still understand the gist of how the proof holds together.
Writing a proof feels very different. You start out with with some assumptions just as before, and you can explore the streams of consequences of each of these premises, however you have no idea which stream you should explore first. Many streams of consequences will lead to dead ends and some consequences you have discovered will only beecome useful until after you have combined them with consequences from the other premises. If all goes well, you end up with the conclusion you wanted and the hardest part is over. At this point you only need to polish the proof. You get rid of all the dead ends that were explored. You structure the proof such that everything flows smoothly. You clarify certain steps you had previously glossed over and fill in any missing details. The finished proof is usually so polished that it gives very little insight as to how the proof was found in the first place.
Despite the thrill of the creative process I have just described, I do sometimes feel that doing Mathematics purely for the sake of Mathematics seems meaningless. If the Mathematics one is exploring is not grounded with some sort of application in mind, it suddenly becomes extremely subjective what mathematical questions are interesting to solve. What I have recently discovered is that mathematicians are split up into small clusters of people and that the important questions to be solved are whatever the community thinks is worth solving. Usually, the senior members of these small communities seem to be the guiding force as to what problems are considered to be interesting, which gives some sense of historical continuity to the different mathematical clusters. I do find it somewhat disturbing that the value of solving some problem in Pure Mathematics depends so much on the subjective value that a mere handful of people in the world place on that problem. Applied Mathematics doesn't have this problem or at least not as acutely since the value depends less on the value judgement of a very small community and a lot more on how useful it is to the many different communities of people who use these mathematical tools for their own purposes.
--henrik Thu Aug 19 7:16:05 2004
henrik - Thu Aug 19 7:25:20 2004
"What is mathematics?" Joyrea
I think that the best definition I have seen is that Mathematics is the study of patterns and structures.
--henrik Thu Aug 19 7:25:20 2004
