So I've been super busy with classes this semester but here's a paper that I recently wrote on the philosophy of mathematics (or the mathematics of philosophy?). It has nothing to do with sustainability, but I hope you fine it entertaining.
The Topology of Epistemology
Topology is the study of connectedness, and of the fundamental properties of things that remain despite deformations. “A topologist is interested in those properties of a thing that, while they are in a sense geometrical, are the most permanent – that will survive distortion and stretching” (Barr, 3). Topologically, a doughnut (torus) and a coffee mug are the same object because their surfaces have the same sort of connectedness. They are continuous with the exception of one hole. If they were made of completely malleable materials you could mold either one into the other without poking new holes or making new connections.
A great many things can be considered from a topological perspective, not just solid objects like tori and mugs. Networks can be seen from the topological perspective, analyzing how the different components or nodes of the network are connected to one another. A favorite interest of mine is the use of topological ideas to better understand ecology. In an ecosystem everything is connected, though most of the connections are indirect. Understanding these connections can help us to understand the ecosystems in a new way. More than predator-prey interactions, even the molecules in the atmosphere and the rocks are connected to an ecosystem through nutrient cycles and respiration. These and others are all ways that topology can be practically applied to understand things in our world, but that’s only one aspect of what topology can do.
Of much greater philosophical interest, you can examine the intersection of topology and epistemology. Epistemology is the philosophical study of knowledge and knowing. What is knowledge? How do we acquire it? How can we know if it is true or false? These are all epistemological questions. “None of us is able to get inside the head of another person, to know what that person knows. Even worse, none of us can step outside our own perceptions to determine what something is “really” like” (Mitchell, 226). This sums up what philosophers refer to as the egocentric predicament. This is also where topology and epistemology have their most interesting intersection. Are we connected to each other? Are we connected to the universe? Do we have a direct connection to absolute reality; can we know what things are “really” like? These are still epistemological questions, but now they are topological questions as well.
Plato had an early conception of the egocentric predicament with his “world of forms”, which was a perfect realm that he described containing the most perfect, ideal versions of everything in the universe. For instance, if you were to want to build a table, you would want to build it to be like the perfect table in the world of forms. The problem here is topological, if the world of forms exists then we certainly have no direct physical connection to it. Try as you might, you can only imitate the perfect table from the world of forms. Our connection in this case is through the mind; if you want the perfect table Plato would tell you to go contemplate “tableness”. He believed that ideas from the world of forms could be revealed by getting people to ask the right questions; in fact he believed that all learning was just “remembering” these perfect forms. Mathematical concepts are certainly from the world of forms, as we can imagine perfect triangles and circles and write beautiful equations to express them with infinite precision; however, whatever we draw or construct in physical reality will only be an approximation of the form. A more modern example of this would be the Klein bottle, the form of which certainly exists and can be expressed, but the physical construction of which is literally impossible to make (though we can get close).
Aristotle believed that we were connected to the world that we find ourselves in. If you wanted to build a perfect table, he would suggest that you go look at a bunch of tables and build one with the best qualities of each, and he might tell you that there is no such thing as a perfect table anyways. Aristotle rejected the world of forms on the basis that it could have no impact on the actual world because no direct connection existed. If the world of forms existed at all, Aristotle might use today’s language to say that it is topologically disconnected from our own.
Of course, Plato and Aristotle came along far before formal topology had been developed, so their worldviews were no doubt hindered by today’s standards. Descartes had a much better understanding of mathematics, being the inventor of analytic geometry, and he made things much more topologically interesting. He set out to see if there was anything at all in his mind of which he could be certain. By melting a piece of fresh beeswax he changed its properties in such a way that he no longer recognized it as the piece of wax, but he knew that it must be the original material. Using this example he established the separation between senses and knowledge. His senses told him that the wax was a different thing after melting, but his mind told him that it was the same piece of wax. It appears that there is a connection of some sort between the mind and the senses, but that the two are not entirely continuous. By establishing the discontinuity between the mind and the senses Descartes called into question our ability to know anything at all from our senses and came to the conclusion that the only thing of which he could be certain was his own existence. “I am, I exist, is necessarily true every time I pronounce it or conceive it in my mind” (Mitchell, 230). This philosophical idea is called solipsism, it is the belief that only my mind exists and everything else is just a perception of that mind. Topologically speaking, only one discrete thing exists in the entire universe, my mind, and there is nothing else to even be connected to.
Descartes worked around this by using the idea of a perfect God who would not deceive him to explain reality. He separated reality into two substances, or realms. The realm of matter is unthinking and extended (it takes up space), and the realm of the mind is thinking and unextended (it occupies no space). This allowed him to escape persecution by the Catholic Church by leaving the soul in their realm while delegating the realm of matter to scientists, but it introduced a significant topological problem, the mind-body problem. If mind and matter are two different fundamental things, how can they possibly be connected? How do they interact? They appear by definition to be completely disconnected from one another. Descartes never addressed these issues.
David Hume did address the mind-body problem, by completely throwing it out. He was a radical skeptic who believed that all knowledge came from impressions of reality and descriptions of the world, which were imperfect and uncertain. The only knowledge that we could be certain of, things like mathematical ideas, could not tell us anything meaningful about the world. Absolute knowledge of reality is forever unobtainable, because our minds are only indirectly connected to the universe through our senses. Bertrand Russell goes further down this path, questioning our senses and maintaining the disconnection of our mind from true reality. We only have our sense data and the mathematical reasoning that we can apply to it. Russell argues that rather than discard all knowledge as uncertain rubbish we should just recognize that we are largely separated from reality and from each other and we should just accept what works as being “true enough”.
Russell’s conclusion seems to be the basis for all of science and mathematics. We generally have a list of assumptions and exceptions to any mathematical process. Pure mathematics does not mirror reality because pure mathematics is itself disconnected from reality. You can prove incredible things in pure mathematics, and you can describe the most amazing objects, but they tell us very little about reality. We take math and make up sets of rules that allow us to apply it to model certain physical situations to some degree of accuracy. We match it to our sense data as best we can and continue to use it as long as it is useful. We are connected to reality only indirectly and can never experience “true” reality. All of our ideas are based on uncertain impressions of the universe that we find ourselves in yet find ourselves disconnected from. The best we can do is to ensure that our worldview is consistent. If the collection of sense data and ideas that we have is continuous and all makes sense, then it is true enough. As long as we continue searching for ultimate truth about reality we will continue to get closer to it, but we must accept that we may never actually attain it.
In Euclidian space it would be difficult if not impossible to produce a picture that adequately expresses our topological relationship to the universe. There is certainly a “neighborhood” with which we can become somewhat familiar, and that is defined by our curiosity and how hard we try to learn something. If I want to go learn about ecology I can go read a book on it, and extend my neighborhood of knowing to the information in that book. The difficulty is that our neighborhoods are not smooth. Due to the limitations placed on us by our relations to our senses it seems that every point of our neighborhood is a boundary point. Nothing from the outside world can ever really be brought within; we can only touch on it. I can go learn more about ecology by reading more about it or performing research, extending my neighborhood to new boundaries, but the first points of contact are still boundary points as well. This is why a picture of our relationship with the world would make little sense. All of the points are boundary points, yet each is connected to our unextended mind. Modeled in 3-space our relation to the universe would be an incredibly complex and somewhat discontinuous surface with every point connected to the point of the internal mind, which could only be expressed in this case as being a 4th dimensional quality of the surface. The human quest for knowledge then becomes a quest to ensure that this surface is as smooth and continuous as possible, by always looking for new observations and discontinuities, and endeavoring to use the tools of mathematics and logic to determine what “is not” and what “probably is”.
Bibliography
Russell, Bertrand. The Problems of Philosophy. Simon and Brown, 2011. Print.
Barr, Stephen. Experiments In Topology. New York: TThomas Y. Crowell Company, 1964. Print.
Mitchell, Helen Buss. Roots Of Wisdom. Wadsworth Pub Co, 2009. Print.
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