The good news is, it’s as done as it’s going to be. The bad news is, I wouldn’t recommend reading it.

According to Jacob Klein, the symbolic character of modern mathematics, and especially of mathematical physics, lies at the heart of the conceptual shift that occurred within science in the seventeenth century (1985, 10)[1]. In adopting generalized symbols as the objects of our mathematical investigations we have acquired an implicit intentionality that was unknown in the ancient world: that not only the *methods* of science, but also the *objects* are of an entirely general character. The results of this shift have now so thoroughly been identified with what we intend by a science of the world that we can hardly see either it or what the alternative might be; an alternative that may only be recovered through a rediscovery of what the ancients intended as the objects of their own science. This symbolic way of approaching the world is not limited to theoretical mathematics, however, but has become the basis of modern physics; what we consider to be the most solid truths about the nature of the world.[2] What does it mean to take a “symbolic art” as the fundamental way in which we understand the natural world, and what is the significance thereof?

First, then, what does Klein mean by this use of “symbolic?” His most concise definition of a symbol is that it is a “second intention” idea interpreted as a “first intention” idea. To clarify, he explains that there are, according to the medieval academics, two basic conceptual levels within which we think. The most direct is the level of “first intentions.” Those are ideas that refer to things. When we use a word for an object, like “tree” or “dog” or “spaceship,” that’s a first intention, because it refers to a thing in the world. In mathematics, a first intention idea would be a number of determinate objects or a determinate geometric shape. One cow, five monads, triangle ABC, and so on (Klein, 1985, 58-63). Klein argues that Greek mathematics always (with the possible exception of book 5 of Euclid’s *Elements*) intended objects of this kind: determinate numbers *of* something or specific figures. In theoretical mathematics the exact mode of being for these objects was open to much dispute, but there was no question that they were definite objects, rather than general ones (Klein, 1965, 123).

We can abstract from the first to the “second intention” by taking an attribute of an object and referring to it, resulting in an idea about an idea about an object. For instance, we can talk about “speed,” “color,” “numerability,” and so on, without referring to particular objects that are fast, red, or in groups of 6. In arithmetic, that distinction is fairly easy to maintain when only operating upon known numbers, but becomes much more difficult in equations with unknown numbers, where the only (or at least the easiest) way to find an answer is to operate upon an unknown number as though it were known, by giving it a name. What is the status of such an unknown: is it of the first or the second intention?

The name for “the number that is not yet known, but will answer this equation” could be *a*, as a kind of abbreviation. In like manner, “the number that, when multiplied by another number and added to a third will equal the unknown *y*” might be named *x*, and so on. Would those uses of *a* or *x* be second intention symbols? In the first instance, it seems that it might, but need not necessarily be symbolic. Not all names for unknown things need be second intention ideas; they might intend a definite object that the speaker simply doesn’t know much about yet. A scientist might say that all matter is composed of very tiny things – the tinniest things in existence, and want a word for this: let’s call them *atoms. *To the extent that he is convinced these very tiny things do in fact exist, the word atoms should be of the first intention, since it is referring to an object in reality, the scientist just doesn’t know much about them yet. Now that there’s a word for them, it will be much easier for him to talk about them, try to figure out what might be discovered about them, and convince others of their existence. Used in that way, *a* is the name of a number of monads that the mathematician does not happen to know yet; when the mathematical problem is solved, however, it will be known. Even if it is never known, if the mathematician is sure that such a number exists, *a* might still be a first intention idea. Thus, *a* could be called an abbreviation or a name for the unknown, rather than a symbol. Diophantus in his *Arithmetic* uses an abbreviation of ς in just that sense. The unknown needs a name in order to be calculated upon, so it is “set out” as S1 (if not multiplied), or S2, S16, and so on depending on how many multiplications of the unknown are needed.[3]

The position of the *x* described above is more tenuous. By further abstraction it is possible to go from meaning by a letter like *a* as an abbreviation of a number that is as yet unknown to the mathematician, to meaning instead any possible number that might be set in the equation. In an equation like *nx+a=y* what kinds of objects are being described? *X* is no longer describing a particular number, as would be the case with 2x+6=12. Instead, *x* has been generalized to mean the name of a second intention idea: *x* is any number of anything quantifiable. *X* and *a* are here numbers that are not only unknown to the mathematician, but are themselves indeterminate.

The oddity is that in algebraic equations we operate on these indeterminate ideas of “any possible number” as though they were themselves numbers. Or, to use Klein’s description of symbolic representation, we interpret the second intention idea as though it were first intention. Thus, Viéte in his books of *Zetetics* begins by assigning a symbolic name not only to the unknown numbers, but to those said to be known as well, so that the answer he derives will be true not only in a particular case, but for any possible numbers that might be assigned. Only at the end are numbers given, by way of example.

Since the seventeenth century we have tended to assume that symbolic mathematics is the natural and necessary way in which to look at all things numerable. There must be a general object to match the generality of scientific method. The symbol is precisely the “general object” of arithmetical operations. Descartes also developed a “general object” for geometry in his coordinate system. However, Klein argues, not only is the treatment of symbolic “general objects” not obvious, but the Greeks, despite centuries of mathematical advancement, never did take that step, but always dealt with determinate numbers *of* something (though in theoretical arithmetic and logistic they were numbers *of* purely intellectual monads) and determinate geometric objects (Klein, 1985, 17).

Finding solutions with symbolic letters is, of course, very powerful because it makes possible the discovery and description of general formulas. It may be, however, that the very power of this symbolism also creates a situation wherein mathematicians, and then scientists who rely upon mathematical formulas, and then the population at large, begin to understand the world symbolically rather than directly.

More interesting is the series of conclusions Klein begins to suggest in his first few *Lectures and Essays*. If modern algebraic formulations are essentially symbolic in the “second intention functioning as a first intention” sense, what does it mean when we then begin to base our physics — the fundamental way in which we understand the world — upon that rather dubious method of generalized abstraction? First, it becomes impossible to represent the findings of mathematical physics accurately in non-algebraic terms (Klein, 1968, p 4; 1985 1-3). The reason for this must be the generalizing of the objects of our study. Physics comes to deal not so much with the being of any particular object, or even with the matter of which the object is made, but with the “laws” common to any possible matter (Klein, 1985, 33). As algebra deals not with concrete monads, or even determinate “numbers” in whatever we might currently mean by that term (including strange things like irrationals), but rather with symbols representing the possible substitution of any number, so mathematical physics presumably deals not so much with particular objects exhibiting particular characteristics or motions, but rather with second intentioned symbolic abstractions representing the laws that govern any possible arrangement of matter, but which at the same time can be operated upon *as though it were* determinate.

Moderns, Klein says, inhabit a world ‘of symbolic unreality (1985, 65), because we take that which is most general, most symbolic, as that which is most known. In experience we encounter a world consisting of solid bodies, permeable liquids and gasses, light, heat, and so on. We observe that these materials have properties that are more or less consistent, and different one from another. While it is possible to abstract matter as such from our experience with material objects, and say that there might be such a thing as matter simply, as when Timeus says that perhaps the demiurge formed the cosmos out of matter that as yet had no form, but it’s impossible to know what such “general matter” might be like. I tend to imagine it as some kind of dense fog, but of course even fog has a good deal of organization to it. Yet modern popular cosmology tends to start with just such an indeterminate mass: in the beginning there was very tightly compressed matter and energy that spread out and organized itself according to various laws; very small charged bits of matter formed themselves around very small and dense bits of matter, and began to circle very fast, because that’s what the law determines when very small charged bits of matter get near one another. Some of these very small constellations of matter were deficient by an electrically charged bit of matter or two, and linked up to share an electron or some such thing between themselves. Order ensued.

Just as it is impossible to discuss modern mathematical discoveries in non-mathematical terms, I keep coming up against a difficulty in interpreting Klein’s thought, insomuch as I do not know any mathematical physics, but only the popularizations and applications of the results. Where, exactly, is the connection between algebra and “symbolic unreality?” What *is* symbolic unreality? Physics teaches us to see the world not as a proportionally ordered cosmos, but rather as a series of processes working themselves out through time. Even the planets came together through laws acting on matter, so that heavier objects attracted lighter ones until solar systems were formed. How is this kind of thing symbolic unreality? It is, of course, far removed from the world as we experience it, and yet people hardly doubt that these laws are at least true in the present, whatever might have been the case in the unimaginably distant past. Klein seems to mean something more or other than that.

In his essay on *Modern Rationalism*[4] Klein suggests that taking the symbolic mathematics and general objects of Descartes mathematics as the primary way in which we understand the world implicitly agrees to his philosophical assertions about the mutual estrangement of mind and the material world as well. His assertions in that respect seem at first to be rather far-fetched. Do “we” really believe all that? The contrast he is trying to draw in that passage is between how we moderns look for the truth of the world through mathematical physics, and how the ancients would look for truth through discourse, and especially through clarifying what people meant by things that they commonly said, but had never tried to reason through very clearly. Klein’s appeal to what “we” implicitly accept seems problematic here just as it did at the beginning of his essay on *The World of Physics and the Natural World* when he declared that “mathematical physics is the foundation of our mental and spiritual life (1985, 3).

Far more compelling is the case for saying that we tend to believe the findings of mathematical physics whether or not we actually adhere to a philosophy that allows us to, as when Klein quoted Sir Arthur Eddington as saying that “there are duplicates of every object about me,” the one an object of the senses; a chair, pen, or table, which is solid and can be trusted to remain solid in one’s hand, and the other a scientific object, composed of space and “electrical charges rushing about with great speed” (1985, 34). That also more closely matches his description of philosophical physicists calling themselves as “Neo-Kantian” or “Machian,” but carrying on in their work as physicists exactly the same, and the same as they would if they hadn’t consciously adhered to a philosophy at all (Klein, 1985, 1). Perhaps it would be more accurate to say that we tend to believe in the world as we experience it and in the world as it is presented to us by mathematical physics in the same way and at the same time, whether or not they are compatible.

There’s a table in front of me that holds things on it, and there are also (presumably) bombs that break apart atoms and could reduce this table to dust for reasons that cannot be explained without resorting to mathematical equations. They must both be true, because somehow symbolically generalized abstractions have been translated into satellites, power plants, computers, cell phones, and so on, and they work. Or so one would be inclined to reason, in a similar way though from a greater distance as one might reason that the table must be solid because things don’t fall through it. Yet the nature of symbolic abstraction makes it very difficult – perhaps impossible – to bring together these two conceptual worlds except through technology. People say of mathematical sciences that they accurately predict what will happen in our experiments, and so they must be true, yet at the same time any attempt to interpret those sciences into non-symbolic speech is *not* true, because it does not accurately represent what the formulas had said. It could not, because in the world there are no “general objects,” but only actual ones.

In summary, symbolic mathematics uses not only a general method, but a general object as well, as though it were a determinate object, as when a symbol for “any number” is operated upon as though it were itself a number. Since Galileo physicists have applied the same symbolic techniques to physical principles as were originally intended as a “general theory of proportions (Book 5 of Euclid), and the “finding of finding” (Vieta’s *Zetetics*). Using these techniques physics has transformed our understanding of the order of the world from an order of proportion to one of “law,” where the only true order is the *way* matter in general behaves when affected by various forces. The full implications of this change are doubtful, but at the least it has created a seemingly unbridgeable gulf between the world in which we live and that we experience with our senses and can understand in imagination, and another world existing in it and yet which cannot be either reconciled with the sensory world or even described in terms other than its own – those of mathematical symbolism. Our willingness to accept this parallelism in what we believe to be true of the world may imply an acceptance, even against what we would prefer or claim to believe, of the separation of mind from world. There is the *true* world we are uncovering through mathematical physics, which can be grasped only with general symbols, and the *apparent* world of solid objects, colors, living beings, and so on, which is produced by underlying laws, but have no separate being and order of their own. Thus, we do not approach the world, even as it appears most familiarly before us, directly, but through mental constructions based upon interpretations of equations.

Is there a way to reconcile the power of modern mathematical physics with our actual experience of the world? Klein suggests that his research on the mathematical intentionality of the ancient Greeks is an attempt of begin at least looking for such an answer.[5] Ancient Greek mathematics as he represents it does not look like an answer modernity could accept, however, because it maintains philosophic cohesion by greatly limiting the power of mathematical undertakings.

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Works Cited

Klein, Jacob. Translated by Eva Brann. *Greek Mathematical Thought and the Origin of Algebra*. 1968 Dover Publications, Inc. New York.

Klein, Jacob. *Lectures and Essays*. 1985 Saint John’s College Press Annapolis, Maryland.

[1] When Klein has the same information in both his *Essays and Lectures* and *Greek Mathematical Thought and the Origin of Algebra* I will usually cite the former, because it is greatly condensed and he usually expresses himself more clearly.

[2] Philosophy, Klein says, “should never forget, even for an instant, that mathematical physics is the foundation of our mental and spiritual life, that we see the world and *ourselves* in this world at first quite ingenuously as mathematical physics has taught us to see it, that the direction, the very manner of our questioning is fixed in advance by mathematical physics, and that even a critical attitude towards mathematical physics does not free us from its dominion. The idea of science intrinsic to mathematical physics determines the basic fact of our contemporary life, namely, our ‘scientific consciousness'” (1985, 3)

[3] Diophantus of Alexandria. *Arithmetic. *Handout from Mr. DuVoison, Saint John’s College.

[4] “[W]hat are these premises of mathematical physics and therefore of all our thinking? … First of all, the science of nature as initiated by Descartes (and parenthetically this would apply also to Galileo) presupposes the distinction between thought and the external world as totally disconnected entities. All efforts to bridge the two and the claim that intelligence is sufficient to grasp the external world (as in mathematical physics) must not make us overlook the fundamental fact that the dichotomy involves a profound distrust of the reality of the world … The fact of supreme importance is that we consider our mind as a mind shut up within its own shell, that we consider our soul as a soul isolated without any possible contact with the outside world. Hence the paradox that the mind that is taken to be all sufficient for understanding the world is preconceived as being entirely dissociated and alienated from the world (Klein, 1985, 57).”

[5] “There may be many ways to overcome this symbolic unreality. One of these ways is to understand how ancient science approached the world” (Klein, 1985, 64).

I wish I had the patience (and time) to read all the material you have. But your summary saved me the effort, being to me a ‘secondary intention’ that communicated Klein’s ‘primary intention.’

If I grasp the concept here (secondary) then I get the impression that ancient mathematics is the wrong site to be digging. The separation between reality and cognitive models seems to me to have existed long before mathematics learned how to apply it.

Logic and rhetoric would be places I would look. What is logic, after all, if not the discipline of thinking about thinking. The data for thinking doesn’t really matter, it’s the reasoning that matters–but that is something intangible as the law of gravity (and as real).

Grammar is similar–words aren’t as important as how they link together.

As for confusing models of the world with the world itself, the sophists were experts, weren’t they? All those logical fallacies I learned about (and which are still so effective) are ways of twisting either language or logic–both of which are models of reality. In other words, as words and grammar map out our understanding of the real world, numbers and mathematical modeling have begun to do the same thing.

Such things are the reason for apophatic theology. Words lead to false images of God, so it is often safer to say what God is not, in order that we don’t confuse our analogies (secondary) for the real thing (primary).

But words are powerful things, leading to other words, and with them new ideas and new words. Likewise mathematical descriptions of nature. Both need to be anchored somewhere–to be sure. No secondary intention is valuable without its primary intention being able to back it up.

Probably, Klein dealt with these things in his works. It was fun for me to think about–thanks for sharing.

“the primary way in which we understand the world implicitly agrees to his philosophical assertions about the mutual estrangement of mind and the material world as well. His assertions in that respect seem at first to be rather far-fetched. Do “we” really believe all that? The contrast he is trying to draw in that passage is between how we moderns look for the truth of the world through mathematical physics, and how the ancients would look for truth through discourse, and especially through clarifying what people meant by things that they commonly said, but had never tried to reason through very clearly”

I like the way you make this contrast clear in your paper. It is much the way the writer of this book I am reading on philosophers and philosophy (The Modern Philosphers Revolution: The Luminosity of Existence by David Walsh) takes the philosopher’s task: to search for truth through discourse and existence. And your paper also makes me realize why St. John’s makes you students stick to your texts. I found a really pretty good article on “scientism” in a journal, which however naturally tends to diminish the thinkers it is writing about because it overgeneralizes their thought and sticks it into a category–scientism in this case. Of course to some extent categorization is natural and helpful but it tends to diminish what may be virtues in a work because the writer has an agenda, well-intended as it may be. So writers may be classified as exitentialists or realists or religious or whatever and the outsider may not realize how very much more there is in a work than the description.

I also like the way you show two parallel understandings of the world to be the case–and how does that make sense? Does it make any sense to say that one is truer than another? We can see how we could end up in a relativistic morass here if we jump to conclusions.

Still leaves me wondering how different numbers can have different properties, and where those properties reside. For example: 2, 4, 6, 8 have the property of being even, whereas 3 and 5 don’t. 9 has the property of being square. You can view these properties by drawing a specific instance of a collection of objects in space (a three by three row of oranges), and it extremely convincing that any collection of nine objects can be arranged in a three by three grid in space, but what actually _is_ the property of being square, or even?

Furthermore, is it truly convincing if we start looking at extremely large numbers? It’s not entirely inconceivable that for sufficiently large numbers these properties might break down – if the number is so big that that number of object could never be put in a square to demonstrate that the number is a square number, then I am relying on the formal axiomatic system that I am using to manipulate the numbers to truly represent the abstraction of number correctly – how do I know this is actually the case?