Sunday, October 18, 2015

Response to Euler: The Master of Us All

In a nutshell, the book Euler: The Master of Us All seeks to detail a number of mathematical questions/scenarios left open that Euler either solved or addressed. William Dunham would begin a section explaining the problem and the previous progress made on it, Euler's particular progress detailed in proofs/mathematical work, and then later contributions, as well as open questions left to future researchers.

The book really highlighted the brilliance of Euler, as well as the seeming absurdity that is his style of analysis. On multiple occasions Euler would simply write an object in terms of an infinite series and then take logarithms of each side, and vice versa. Euler techniques were purely for the purpose of analysis, lacking the formalism of proof writing. However, one could argue that this let Euler be more creative as he simply pursued mathematics in terms of reaching results, without regards to crippling  cases and full comprehensiveness. On occasion Euler can be faulted for his lack of rigor, but then again one can hardly imagine the number of successes if he restricted his particular style. As the author stated, "One could reasonably ask whether modern mathematics would even exist without him." Euler also loved to tackle the same problem from multiple directions in order to solidify the strange results he derived to his doubters, as well as test his understanding and prowess.

Another observation I made is the sheer volume, as well as variety, of Euler's contributions. His work ranged from geometry, to number theory, to infinite series, and to complex analysis; this is an incredible body of work for one man. Interestingly enough, every area he brought his unprecedented analytical abilities and insight, even areas such as geometry where proofs were usually performed using clever insights rather than analytical techniques. His willingness to examine old topics in new ways (e.g. the relationship between the centroid, circumcenter, and orthocenter of a triangle) as well as work with new topics that others refused to consider (e.g. complex numbers) is simple incredible.

Although not comprehensive, this work gives the reader enough material to come to an appreciation of Euler and his works. Although Euler may not be the master of us all, he was certainly a master of his craft and was a person is not easily equaled.

On the Axiomatic Nature of Logic Systems

Although in a previous blog post I have already addressed the relationship between math and science, both being simply human creations to understand and predict various phenomena, I will take this blog post to instead further highlight the similarities of both as well as respond to some of the criticism (mostly self-inflicted) of the previous post.

To understand the studies of math and science we must start at the core of each. Mathematics must start somewhere, this much we know. Meaning the entire field of mathematics concerns itself with proving new results from old ones, so we must eventually assume that something is intially true. Of course such a statement is called an axiom. An important note about axioms is although we cannot prove them, we do not believe they are true for no good reason. We can perform operations (e.g. A+B) and make observations for what we believe is true (A+B=B+A since the order should not matter). For instance, saying that there is an integer 0 such that a+0=a for all integers a matches physical observation (I can add nothing as much as I want to a pile but the pile remains the same.)

Therefore we should ask if science have axioms, and if so, what are they? To see scientific axioms we must delve briefly into questions that philosophers, not scientists, ask. For example, how do I know the world I perceive is real? Meaning how do I know that this reality is real, not some weird pseudo-reality (think The Matrix) or that reality doesn't just exist in my own head. Despite the "proofs" of one's existence (e.g. I think therefore I am- Descartes) one simple cannot prove that one exists insofar as to be able to reliably trust one's senses to the extent of being able to make accurate scientific conjecture (and I believe that this is an inherently unsolvable problem). This inability to answer the most basic metaphysical question has plagued much of philosophy for millennium. So how do scientists make claims about how the universe functions? The simple answer is that they just assume basic properties of reality: that 1.) the universe is real to point of understanding and measuring it and 2.) we are able to measure physical properties of the universe. Again these are reasonable axioms, since by taking them as true we are able to predict physical phenomena and build neat devices utilizing this knowledge (e.g. we can build the computer/phone/tablet you're using to read this.)

So we come to the conclusion that both math and science are inherently axiomatic (and I conjecture that all logic systems share this property.) Unlike my previous post where I claimed that science and math were inherently different since the physical and mathematical worlds are inherently different, the reality is that the existence of each world is dependent on humans' belief in them. In other words, since I cannot prove the existence of the physical world and therefore simply assume it, it does not exist without this assumption. We then reach the conclusion of the previous post while covering up some of the logical inconsistencies present in the original theory.

Note: We can think to apply the same ideas to other logic systems: religion, music, cooking, etc. For example in the case of religion one assumes the existence/non-existence of a overarching reality (e.g. God) based on observation.