Math, Subjectivity, and Truth

A week ago, I might have said something like the following: When I think of computers, I think of math machines. When I think of math, I think of something that is complex, frustrating, and infallibly true – in my mind the study of mathematics, unlike what I do as a humanist, has right and wrong answers. It is objective. On some level, that might make math “better,” or at least more valuable and useful. On another level, that makes it less fun.

Reading Martin Davis’s The Universal Computer has significantly changed the way that I understand both math and computers and in the process has made me feel somewhat foolish for my initial views. The Universal Computer follows the lives and discoveries of mathematicians whose theories contributed to the creation of the computer. One of the main focuses of the book is the way that logic, and different types of logical thinking, led to the creation of the computer. Davis makes it clear that it was not math, or at least not the type of math I learned in high school, that was instrumental in bringing about the computer. Rather, it was a quest for logic, language, and truth that developed over the course of centuries before leading Alan Turing to envision the universal computer. Davis notes that Liebniz, who paved the way for a binary numeral system, “would seek a special alphabet whose elements represented not sounds, but concepts…a true alphabet of human thought.” Liebniz’s goals make me rethink the popular dichotomy between CS and the humanities. Truth, logic, and language are also very much at the root of what humanists do.

One of the themes that emerge in Davis’s narrative is that the logical principles that arose over the last several hundred years were often points of academic contention. I guess I’ve never really thought of the evolution of math before. I have always known that other disciplines, like biology and physics, have progressed, but on some level I assumed that mathematical truths are so intrinsic that they have always been the same. Far from treating the current state of mathematics or the invention of the computer as inevitable outcomes, Davis’s narrative gave me the impression that at many points along the way things could have gone differently. For example, Davis contemplates the type of progress Liebniz would have been able to achieve under different personal circumstances. The fact that each person that Davis writes about is contextualized with historical events and personal details makes the book more interesting, but even more importantly, it points to the way in which external factors influenced the creation of the technologies that dominate our lives. I think that’s really interesting. It also makes me reassess the way I associate objectivity with math and computer science.

The issue of objectivity, and how it relates to different disciplines reminds me of a conversation I had this week with a few of my students. They were anxious about the grades they were going to receive on their first paper. This led us to discuss how grading writing is subjective. However, I was surprised to find that my students thought that the grading in their science and math classes was not subjective. But who determines what’s on your test? I asked. Who determines whether or not you get partial credit for “thinking” about the problem in the right way? Who decides the best approach to solving a problem? I think this ties back to Davis in that some questions do have a right and wrong answer, however, determining a question’s phrasing, or even determining if it is the best question to ask, is often subjective.

I think one of the most fascinating examples of this type of subjectivity can be found in Davis’s description of two competing approaches to the creation of the universal computer – Von Neumann’s reliance on hardware versus Turing’s minimal approach and reliance on software. Neither approach is “wrong” in the sense that it doesn’t produce a true or viable outcome. However, each approach has radically different implications for both programmer and user. I think that the differences in Neumann and Turing’s approaches (as well as the differences between say Hilbert and Kronecker) are something worth thinking about. It’s easy to fall into the type of logic that says computers are the way computers are because they couldn’t be any other way. The Universal Computer reminded me that it is a humanist’s job to question basic assumptions, preconceptions, and the insidious types of processes and logics that at first glance seem inherent.

4 thoughts on “Math, Subjectivity, and Truth

  1. One of the many thoughts triggered by the readings was in my case to question language and subjectivity. The representation of symbolic logic made me thought of those primary states of language —maybe those that exist only in the theory but never existed besides a long and gradual process. Apart from my lack of preparation to follow some of the explanations on the book, I also had a little outburst of confidence and saw the potential in all of us —literature students. We actually do manage a symbolic system, the most widespread in the world and what it is probably the one to blame for the biggest culture transmission. That self encouraging thought came to me after the simple redaction of a SMS, where the little space provided exposes the importance of the layout of characters and the position of every expression we choose to communicate. Language is a highly subjective system too, but we have forgot about its arbitrariness and lack of intrinsic meaning.

  2. I like the way you connect the actual history of the computer with the way we might approach it in the present; I think it’s absolutely true that the shared focus on language and truth can allow us, as humanists, a ‘way in’ to what might otherwise seem inaccessible. I would even take it a step further and say that all the contention, all the what-ifs that Davis tracks, can act as footholds for us going forward, as we endeavor to influence the development of technology, and as we try to marry it more with our own particular areas of interest. The freedom to question what previously seemed objective is something that gave us the computer as we know it, and is also what will give us the logical approaches of tomorrow.

  3. Your discussion of Turing and von Neumann’s architecture points to a theme I think you elucidate nicely in Davis’ book: namely, the ways in which social forces (both on an individual and meta level) shaped the development of the computer. In the case of von Neumann’s architecture, it seems to me that it has been widely adopted because it mimics epistemological forms that we are familiar with. A central processor, RAM and hard drive is legible as the way that we conceive of memory: analytics, short and long term memory. RISC computing doesn’t work so well in that framework, and thus we have technologically chosen not to pursue it. It’s not about technical merit. It’s about human choice.

  4. When I was about six months younger, I liked to rib a mathematician friend of mine that all he was doing was outlined by the Port Royal Grammar in the 17th century.

    But fundamentally, this is a great insight Gabi. Mathematics, especially logic, is linguistic. And how one thinks about the language, whether Turing or von Neumann or Cantor or, matters for creating artificial languages.

    As you say, the distinction between CS and the humanities isn’t always what is seems!

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