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Math According to Archimedes and Hardy

February 1, 2019

Thanks to Alissa Simon, HMU Tutor, for today’s post.

I have a number of questions still rumbling around after Harrison Middleton University’s January Quarterly Discussion. We read Archimedes’ Sand Reckoner and G. H. Hardy’s Mathematician’s Apology. I put these two pieces together because I am interested in mathematical discourse separated by thousands of years. More than time, however, they also came from different parts of the world, encountered very different technological advances, and lived immensely different lifestyles. Archimedes of Syracuse was a Greek mathematician and inventor who lived around 287-212 BC. Hardy, on the other hand, was born in 1877 in England and showed an early aptitude for numbers. He continued with math through college when he became largely interested in “pure mathematics” which, he claimed, is more noble than practical math. So, my first question is whether or not Archimedes’ Sand Reckoner corresponds to pure math, or practical math?

In The Sand Reckoner (which I have written about before), Archimedes sets out to demonstrate that math has strategies to break down something as large and abstract as the measure of the universe, or the grains of sand on earth. His proof begins with rather large assumptions, such as “I suppose the diameter of the sun to be about 30 times that of the moon and not greater.” Initially, I did not understand why Archimedes would base a proof upon such unknowns. However, I have always thought that the exercise was more to inspire imagination than prove an actuality. And now, based upon conversation during the Quarterly Discussion, I see that Archimedes wanted not just to inspire imagination, but to demonstrate the potential of math. He was explaining that math functions on strategies which engenders new information. This would be important, of course, living in a time when math was largely unknown and therefore, seen as untrustworthy. So, to me, The Sand Reckoner is not a proof of any one thing, but a proof of math itself. He asks his king, other educators, and perhaps his community to believe in the potential of math and to contemplate questions of great size.

Jumping forward to Hardy’s piece, then, he draws a very decisive line between practical mathematicians and pure mathematicians. Practical math builds things like bridges and steam engines. Pure math contemplates greatness. For some reason, Hardy’s differentiation always brings me back to Archimedes, who built levers and invented all sorts of practical things, but yet also contemplated the universe. Does the mathematician who builds the bridge not also dwell upon other possibilities? Surely not all of them do, but I find Hardy’s approach very severe and limiting. I am not sure if his words are meant to inspire others to attempt a career in math, or to explain to the masses how little they actually know. Either way, I feel that the work fails when placed next to something like Archimedes’ proof which shows math’s potential rather than belabors the value of ambitious men. Perhaps, though, my perspective is naive, since I do not grasp much of the math that would place me in this elite group.

Clearly Hardy values creative thought over any other pursuit. I can identify with this, but I wonder if his criticisms speak to moral dilemmas of his day. Hardy wrote A Mathematician’s Apology in 1940. I have to think that war-time inventions must have been on his mind when he differentiated between practical and pure mathematics. And yet again, I return to thinking about Archimedes who built many machines of war such as the Archimedes Claw and catapults. Does this remove him from the rank of pure mathematician (if he was ever considered such)? In theory, I believe that I understand Hardy’s point. In fact, I relish the idea that a life of creative thought or philosophical discourse is as worthy as shipbuilding. This would justify my own life as well. However, it seems rarer that society allows such thinking to exist. Rather, society is structured in a way in which we must all pay for food and shelter, and creative thought does not pay. I think that perhaps Hardy might have been trying to tell us, the public, that we should value creativity more than we currently do.

Additionally, his message does not address morality at all, which the group found interesting. I wonder how Hardy would tie ambition to morality. He glories in the uselessness of math because it cannot be tied to evil. He writes,

“If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither [Carl Friedrich] Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in times of war); and both Gauss and lesser mathematicians may be justified in rejoicing that there is one science, at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.”

According to Hardy, pure math never filters into practical applications. I find this reasoning illogical, though since again, levers as created by Archimedes were once thought impossible and are now the foundation of much greater machines. In my mind, the lever was purely theoretical at one point and is now elementary science. Also, once public, how can anyone protect the ways in which their work will be used (or not used)? How can Hardy surmise that the pure math of today will not be the applied math of tomorrow? And does its application make it any less pure?

As always, I am indebted to a wonderful group who wanders through these questions with me. The next Quarterly Discussion will be held in April 2019. For more information email asimon@hmu.edu. I look forward to hearing from you!

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Bohr's Use of Language

December 14, 2018

Thanks to Alissa Simon, HMU Tutor, for today’s blog.

At the end of the fourth chapter of Atomic Theory and the Description of Nature, Niels Bohr writes, “Besides, the fact that consciousness, as we know it, is inseparably connected with life ought to prepare us for finding that the very problem of the distinction between the living and the dead escapes comprehension in the ordinary sense of the word. That a physicist touches upon such questions may perhaps be excused on the ground that the new situation in physics has so forcibly reminded us of the old truth that we are both onlookers and actors in the great drama of existence.” I love the stage analogy that Bohr uses. I picture a camera forever panning backwards. When the scene begins, we are looking at a stage, but as the camera moves backward the audience is on the stage. Included in my visualization is that both the stage and ourselves become increasingly smaller. This is important to the way that I see Bohr’s argument. Bohr suggests that even if we can claim to know pieces of the whole, we will never see the complete picture at one time. This is not to say that we cannot connect pieces in the way that we do a puzzle, but that no single piece can stand as significant of the whole. Atomic Theory and the Description of Nature explains that the future of science will be (and already is) beyond our senses. Instead of seeing reactions and experiments, we must rely upon a variety of tests, the accumulation of which will grant a picture of the whole. At no one time, Bohr reminds us, will we be able to actually see the whole, however. In both this piece and in “Discussion with Einstein on Epistemological Problems in Atomic Physics,” Bohr explains how his view differs from Einstein. Unlike Bohr, Einstein believed that at some point we will have a complete picture of atomic physics.

A recent discussion of these readings sparked my curiosity about the things which validate science, such as observable data. I am also interested in the way that Bohr compares atomic theory to classical philosophy. By this, I mean that he understands that there are unknowns in atomic theory. Finally, I also want to know more about the way he emphasizes that the scientist is a part of the experiment. In Atomic Theory he writes, “The resignation as regards visualization and causality, to which we are thus forced in our description of atomic phenomena, might well be regarded as a frustration of the hopes which formed the starting-point of the atomic conceptions. Nevertheless, from the present standpoint of the atomic theory, we must consider this very renunciation as an essential advance in our understanding. Indeed, there is no question of a failure of the general fundamental principles of science within the domain where we could justly expect them to apply. The discovery of the quantum of action shows us, in fact, not only the natural limitation of classical physics, but, by throwing a new light upon the old philosophical problem of the objective existences of phenomena independently of our own observations, confronts us with a situation hitherto unknown in natural science. As we have seen, any observation necessitates an interference with the course of the phenomena, which is of such a nature that it deprives us of the foundation underlying the causal mode of description.” As with classical philosophy, we are at a crossroads. This new path is filled with unknowns, and not only that, but unobservable unknowns. Despite this complication, Bohr asks scientists to depend upon established terms which maintain a sense of cohesiveness, but also give us some concrete foundations for theoretical science. This technique hearkens back to the beginnings of philosophy as humans grappled to find language suitable for metaphysics.

The “old philosophical problem of the objective existences” outside of our own hearkens back to the roots of philosophy. In fact, as science moves forward, it must address many of the same questions that began as early as 2000 years ago. To address some of these unknowns, Bohr demands precise language without straying from classical vocabulary. Both Atomic Theory and “Discussion with Einstein” address the difficulty of language for the scientist and for the public. He explains that unknowns do not equal a lack of knowledge or a scientist’s uncertainty about the validity of their research. Rather, an unknown is in itself useful. He labels this dilemma an “intricacy of language.” Bohr writes, “[Q]uantum theory presents us with a novel situation in physical science, but attention was called to the very close analogy with the situation as regards analysis and synthesis of experience, which we meet in many other fields of human knowledge and interest. As is well known, many of the difficulties in psychology originate in the different placing of the separation lines between object and subject in the analysis of various aspects of physical experience. Actually words like ‘thoughts’ and ‘sentiments,’ equally indispensable to illustrate the variety and scope of conscious life, are used in a similar complementary way as are space-time co-ordination and dynamical conservation laws in atomic physics. A precise formulation of such analogies involves, of course, intricacies of terminology, and the writer’s position is perhaps best indicated in a passage in the article, hinting at the mutually exclusive relationship which will always exist between the practical use of any word and attempts at its strict definition.” The imprecision in language exists in all fields, and grows as the field grows. Bohr’s insistence upon utilizing classical terminology is twofold. First, He asks that we use exact, well-defined terms so as to limit misunderstandings. Second, he wishes to avoid further abstraction of an already abstract subject.

Bohr’s focus on the language debate reminded me of a recent article on modal verbs, or verbs which predict rather than describe simple facts. The article claimed that scientific papers often get buried or dismissed because they include words such as “might,” “could,” “may,” “ought,” or “will.” Of course, these verbs reflect the fact that scientists do not have all the answers, and each experiment leads to further unknowns. This dismissal is something that Bohr feared and a reason for his insistence upon classical terminology. Incorporating existing terminology with atomic physics, science remains valid and as independent of the scientist as possible. Again, I am reminded of the fact that, according to Bohr, the scientist is a part of the experiment as much as they are observers. Therefore, if the scientist were to also alter terminology in a way that best suits their vision, they would further insert themselves and their view into the experiment. Furthermore, modal verbs signify opportunity for further experiment. They also reflect Bohr’s insistence upon the fact that we cannot know the whole picture anymore. As we interact with and learn from the world, the complexities in science grow larger. However, while uncertainty can be off-putting, uncertainty in science should be celebrated.

Bohr’s focus on language makes me think that there are opportunities for educators here too. In teaching science (to both scientists and non-scientists), we should include a better understanding of the specificity of language. We can also explain the benefit of things like modal verbs. Perhaps this will better enable us navigate complicated theories and unobservable data. We could also better educate young scientists with writing skills. Integration of these fields seems inextricably tied together. Bohr speaks of the writer’s dilemma which he calls, “the mutually exclusive relationship which will always exist between the practical use of any word and attempts at its strict definition.” In some senses, the scientist is now also a writer. In other words, language is of extreme importance for the future of science and we would do well to also teach according to these principles.

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Socrates: A Sophist?

October 26, 2018

Thanks to James Keller, a 2018 Harrison Middleton University Fellow in Ideas, for today’s post.

With his head in the clouds, Socrates, as portrayed by Aristophanes, is a figure of mockery. Not only that—he is a sophist. One who comes to The Clouds only after reading the Platonic dialogues may be startled at this discovery. He may ask, Are we even talking about the same person? That Aristophanes considers Socrates to be a sophist is most shocking. Certainly, public figures are often subjected to mockery, and though Socrates has been a celebrated thinker after his death, he was not so celebrated in life. But that he should be considered a sophist? Unthinkable. It is almost inconceivable that Plato, who in The Sophist considers the sophist to be something of an anti-philosopher, should have studied with and revered a sophist. Moreover, the Socrates that appears in Plato’s dialogues is pitted against the sophists, particularly in Protagoras, Euthydemus, and Gorgias. How is it, then, that Aristophanes could think that Socrates was himself just another sophist? Yet, Aristophanes’ perception may not be inexplicable when one notes the similarities between Socrates and the sophists as they appear in Plato’s dialogues.

The Socrates of Plato’s dialogues is most renowned for his method of inquiry, Socratic questioning. In order to test the wisdom of certain figures and in order to clarify his own ideas, Socrates asked his interlocutors a series of questions, a particular form of dialectic. Despite its name, however, it is quite likely that this was not his invention. Plato gives no indication that this form of questioning was unique to Socrates even though other characters express exasperation at his questioning. Indeed, characters other than Socrates use the same method or one quite similar. In Euthydemus, the sophist brothers Euthydemus and Dionysodorus also employ questions as part of the dialectic process, a practice that appears natural to them. And, in one of the later dialogues, a young Socrates does not ask the questions but receives those given by Parmenides, after whom the dialogue is named. This suggests that what is called Socratic questioning actually precedes him and was a tool of sophists. To an outsider, contemporaneous with Socrates, it might then appear that Socrates’ disputes with the sophists was not a repudiation of sophistry but an inter-sophistical dispute.

Nor might his method be the only perceived similarity between Socrates and the sophists. In Plato’s portrayal of the sophists, the sophists crave acclaim. Applause punctuates their arguments and speeches in Euthydemus and Protagoras. They love an audience and they love playing to an audience. Socrates can be contrasted to them in that he does not seek the approval of an audience, not in Plato’s version of him anyway. Nevertheless, he does gather an audience. Various characters do root him on in the dialogues. And in The Apology, Socrates mentions that young men like to follow him around for the sake of being amused. As he roams through Athens challenging various authorities to prove that they actually do possess the wisdom they profess, he proves them to be lacking. This act of revealing authorities to be fools—or, if not fools, pretenders to expertise that they do not in actuality possess—is unsurprisingly found to be entertaining by some. To an outsider, it might look like Socrates was trying to make a name for himself, just like a sophist might.

The source of this amusement was different, but even that might look the same to an outsider, especially one who only knew Socrates by reputation. Euthydemus and his brother also make fools of others, but that is because they build absurd arguments that make their interlocutor appear to have said something foolish. It is as if they tricked him. They treat argument as a sport, playing word games to prove such absurdities as that a man’s dog is his father. They are facetious and mocking, and they leave their interlocutors frustrated and sputtering, fearing to answer lest that answer be twisted and used against them. Socrates may have shared a similar reputation, as he also left his interlocutors speechless. In Meno he describes himself as a torpedo fish that leaves others stunned. But an important difference separates him from Euthydemus and Dionysodorus. He is not playing word games; he is looking for clarity. He asks people to define terms that they take for granted, and to their great consternation, they often discover that they cannot. A well-known example of this appears in Euthyphro where Socrates leads the eponymous priest to the realization that he cannot properly define piety. After discussing the question for some time with Socrates, the priest hurries away, uncomfortable with the conversation. But never did Socrates play a linguistic trick upon Euthyphro. Never did he seize on an ambiguity in language to make a fool of the priest, turning the conversation to mere jokes.

Many of Plato’s Socratic dialogues end unresolved, which speaks to another difference between Socrates and the sophists. As represented by Plato, the sophist teaches others how to win arguments, unconcerned with whether the argument is correct or not. (See, for example, Gorgias.) Whatever the point is to be argued, the sophist will be able to prove its truth. But Socrates’ goal is not to win an argument. He desires to find the truth. The sophist asks leading questions in order to get an admission from his interlocutor. Socrates uses questions to better understand the arguments of others, to challenge them—yes—but not necessarily to overthrow them. It is the truth he is after, not victory. Argument is not a contest to him, but a means for inquiry. So, at the end of a dialogue, Plato does not show Socrates on the field of verbal battle having won the day and turned back all comers. Socrates is much more likely at the end of a dialogue to announce that, though no answer has been discovered to the question being discussed, still he and the interlocutor must not stop seeking after the truth.

To an outsider, perhaps it would appear that Socrates was just another sophist, asking endless questions to make fools of others, seeking fame, and winning an argument at all costs. Perhaps, he even started out that way, first learning with sophists and only later going his own way. But the similarities between Socrates and the sophists is ultimately superficial. Socrates, at least as portrayed by Plato, was not concerned with winning arguments at all costs. He would have seen that as a truly pyrrhic victory. He used the same methods as the sophists to achieve a different end: truth. In this way, Socratic questioning is properly named after him, because he used it for shared inquiry, not to lead others into verbal traps. If Plato’s portrayal of Socrates was closer to the truth, it is a tragedy that the comedian Aristophanes did not see it.

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Language Games

October 12, 2018

Thanks to Alissa Simon, HMU Tutor, for today’s post.

Communication is awfully complicated. How does anyone know, for certain, when they are communicating? For meaning to occur, two parties must have some knowledge in common. Ludwig Wittgenstein wrote many pages about the way that language is structured. Today, I want to investigate his idea of the language game and then apply it to Heidegger’s idea of Being.

According to Wittgenstein, the language game begins with, but does not include, names. He refers to the action of naming as “preparation for description” (329B). That a name for something exists only means that we have a shell of reference. So, I can mention a cat, which will give you a categorical reference devoid of specifics. Once we have assembled some names, we begin a discussion by adding descriptors. Wittgenstein likens this to a chess board. Names are the pieces that we can move around the board, but they are not the game itself. Now that we have these categories, we can begin to communicate about them, describe them, fill them in, move them. Wittgenstein writes, “[A] great deal of stage-setting in the language is presupposed if the mere act of naming is to make sense. And when we speak of someone’s having given a name to pain, what is presupposed is the existence of the grammar of the word ‘pain’’ it shews the post where the new word is stationed” (Philosophical Investigation #257). So, the language game takes concepts and places them within a structure.

The knowledge of concepts, however, is of crucial importance. Wittgenstein continually warns the reader that meaning is not a given. In example after example, Wittgenstein describes how difficult it actually is to make meaning. He writes, “[I]t is difficult to see that what is at issue is the fixing of concepts…. A concept forces itself onto one” (425B). What he intends here, I believe, is that the concept itself has been defined by culture, society, norms, etc. In the chess analogy, the knight’s movement has been defined for you. You can only move it in an ‘L’ shape according to the rules of the game. Say, for example, that your language game intends to discuss the idea of a cat, “cat” will already have an agreed-upon definition. This concept, however, is fixed only in terms of this specific game. Once you exit the game, cat may contain more or less meanings, more or less descriptions. Meaning, then, depends upon the group involved in a single discussion as well as the terminology that the discussion utilizes.

Furthermore, Wittgenstein discusses anomalies, such as mistakes, calculations, guesses, hypotheses, etc. Upon what foundation do we make a mistake? Is it fair to call a lion a cat? Though it fits the category, it may not actually represent the idea or concept driving the speech-act. For instance, if I make the statement: “The cat is cute,” in what sense would lion make sense and in what sense would it not?

Now that we have a basic idea of the language game, we can move from Wittgenstein’s Philosophical Investigations into Heidegger’s “What is Metaphysics?” Near the end of this piece, Heidegger claims:

“Obedient to the voice of Being, thought seeks the Word through which the truth of Being may be expressed. Only when the language of historical man is born of the Word does it ring true. But if it does ring true, then the testimony of the soundless voice of hidden springs lures it ever on. The thought of Being guards the Word and fulfils its function in such guardianship, namely care for the use of language. Out of long-guarded speechlessness and the careful clarification of the field thus cleared, comes the utterance of the thinker. Of like origin is the naming of the poet. But since like is only like insofar as difference allows, and since poetry and thinking are most purely alike in their care of the word, the two things are at the same time at opposite poles in their essence. The thinker utters Being. The poet names what is holy.” (310B)

This passage strikes me as thought-provoking (and complicated) for many reasons. Heidegger mentions a cleared field, which is an important aspect behind his idea of essential Being and Word. This field is, in fact, a Nothing through which we come to understand Being itself. If we think of the cleared field as a field of possibility, we are able to project our Being into it. And then, Being(s) exist because we do. According to Heidegger, this constant process of understanding the world through a removal of everything is the first step in thinking. Heidegger writes, “Being is not a product of thinking. It is more likely that essential thinking is an occurrence of Being” (309A). In other words, once the field is cleared, a Being can focus on a field which allows for contemplation of a thing or things, but not everything simultaneously. He asks that we focus on the Word, meaning a specific idea devoid of self and other baggage. From there, we will find thought.

The final line of his long quote above mentions the difference between a poet and a philosopher. Basically, according to Heidegger, they both work toward the same goal. However, the poet stands at one end of this spectrum while the philosopher at the other. The difference arises in the mode of expression. So, the philosopher seeks a discursive, direct expression of thought, whereas the poet seeks truth through metaphor. In other words, the poet attempts to fully remove Being itself, and focus on the thought, focus on embodiment of the other. In this way, the poet arrives at a similar, but different, idea of the moon (for example), or whatever body you would like. For this reason, Heidegger claims that the philosopher arrives at an understanding of Being, whereas the poet finds what is holy.

Much remains unpacked in this short commentary on Wittgenstein and Heidegger. However, we have arrived at an idea of Being as represented by Heidegger’s very specific terminology. Heidegger is known for co-opting or creating words and phrases for his own purpose, devoid of their everyday meaning. In some cases, these phrases are untranslatable (as we find in the passages regarding Da Sien). That does not mean, however, that nothing can be gained. In fact, I hope this short experiment has granted some window of insight into a discussion of language itself.

*All citations are from the Great Books Anthology number 55, 20th Century Philosophy and Religion, 1990.


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