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Democratizing eighth-grade algebra promotes social justice. (Brookings Institution)

Money, mechanization, algebra. The three monsters of contemporary civilization. Complete analogy. (Simone Weil)

Mt. Airy, Philadelphia. There are a lot of conspiracy theories out there, but odds are that not many raise alarms about the fact that students in America and elsewhere are taught algebra before geometry. Yet this may be one of the more potent means by which the vision elaborated by Descartes works its way into the very foundations of how we imagine the natural world.

We are all familiar with the coordinate system devised by Descartes (or rather modified from the one devised by Descartes) – the x axis and y axis that enable us to describe the location of any point in a plane. Add the z axis for 3-D plotting, and we have a way of imagining and quantifying space abstractly. This kind of analysis of spatial magnitude provides the basis for the exact measurements of motion at the foundation of modern physics. Since modern physics provides most of us with our basic understanding of the character of reality or “nature”, it is fair to say that this understanding, in its imaginative dimension, is largely a Cartesian picture. And since most students who make it through high school get the Cartesian picture of mathematical space worked deeply into their imaginations (whereas very few are sufficiently imbued with a grasp of physics to have their imaginations deeply formed by it), it is fair to say that it is by means of coordinate-plotted geometric algebra that our imagination of the physical world gets shaped.

At the basis of this new analytic geometry is a profound transformation in what I will call the metaphysical imagination. (While crucial aspects of this transformation were accomplished by Francois Viete half a century before Descartes, it is still Descartes who elaborates most completely both the stance toward the world of appearances that enacts this new metaphysical sensibility and the conceptual and mathematical tools that direct the pursuit of the kind of knowledge that corresponds to it.) This transformation has everything to do with how we understand number.

Consider what the coordinate axes tell us. Each of them is a “number line.” That is to say, between any two numbers on the x axis, there will always be another number; and if we imagine taking this process to the limit, what we approach is a continuum of numbers. There is no fundamental difference between number and extension in length, or, in other words, between multitude and magnitude.

For the classical tradition, there is an important distinction between multitude (number) and magnitude (extension). Number or multitude is discrete; extension or magnitude is continuous. A number, according to Euclid, is a “multitude of units.” We call these the “natural numbers” (justly so), and consider them one subset of a more inclusive set of numbers that encompasses others types, like rational and irrational, real and imaginary, positive and negative. For the classical tradition, the “natural numbers” are the numbers period (though when it comes to computation, there is some difficulty figuring out what to call fractions). This is because when we speak of a number, we are ultimately referring to a multitude of things. Beings that can be counted exist in nature as unities.

For the ancients, the problem of number leads directly and explicitly to metaphysics. The basic question is how it is possible for there to be two of something. They can only be two of something if that something is the same; but they can only be two of that something if they are different. And of course this leads us back to the question of how each of these things is able to be a unity in the first place. And this leads to identifying form as the principle of unity for each being.

Form is most evident to us in the bodily shape of a thing. This bodily shape is as it is because of the way that body is organized and articulated. But in the case of a living being, the body has this organization and articulation because it serves as the material basis for the various capacities and life-activities that are characteristic of that living being and constitute what it is to be such a being. The living being’s self-sustained complex of capacities for its array of characteristic life-activities is what Aristotle calls its form and its soul, or the governing principle of the living being that it continues to be. Thus the form that appears to my senses gives me access to the form that is the very being of the living thing itself.

It is number, then, and the question of the natural unity and integrity of natural beings, that guides our understanding of nature for the ancients. Extended magnitude is a secondary characteristic of beings, incident to their material aspect. Form is the more fundamental aspect of its being, because it is the form that dictates what material (and how much of it and in what proportions) will be present in the living being or artifact.

Descartes changes this. When he develops his algebraic geometry, he chooses to represent numbers by lines, collapsing the distinction between magnitude and multitude. The emphasis is on quantitative relationships, and what we take as our unit is arbitrary. When he analyzes the basic characteristic of the being of the “external” world, he finds it in extension, not in the presence of given unities. As a result, we have a science that examines processes rather than beings, for a process is a change over time of the relationships among various measurable magnitudes. And we have the analysis of those beings into their simplest parts, the parts most easily described in purely algebraic-quantitative terms. We lose a sense of the integrity of beings as active wholes, a formal integrity that makes them what they are. Thus Dawkins, in insisting that the organism is not a unity but rather a “colony of genes”, is being a genuinely Cartesian biological theorist.

But it is not only the integrity of natural beings that is lost; it is also a sense of place. We imagine the world as abstract space. Quantities of matter are present in parts of that space, and can be moved to other parts of that space. It’s just a change of coordinates. The x and y axis meet at the “origin” of the coordinate system, but that origin itself is arbitrarily placed. All space, as space, is interchangeable.

In classical Greek, in Aristotle’s physics, there is no word for “space”, only for “place”. That is certainly not because Aristotle didn’t know his geometry. On the contrary, it is because he knew it deeply, and knew its difference from his arithmetic. He knew that the world is a world of beings actively maintaining their unity and integrity in their places, and that geometry is an abstract representation of a limited aspect of things in that world, and indeed more abstract than arithmetic because arithmetic deals in wholes.

In Euclid’s Geometry, two of the thirteen books deal with number. He proves many theorems about proportions among numbers that he then proves all over again for continuous magnitudes in the subsequent books. To the modern mind, trained by Descartes’ geometrical algebra to collapse this distinction, this seems terribly inefficient. But Euclid seems to have thought it important to maintain this clear distinction. It is a matter of fidelity to the nature of reality rather than to sheer computational power and convenience.

The rest of Euclid’s geometry is perfectly intelligible independently of numerical expression. There is a marvelous beauty to the way the relations of figures are revealed when Euclid proves that the area of the square on the hypotenuse of the right triangle is the sum of the areas of the squares drawn on the other two legs. This beauty is entirely missing in the formula “c squared equals a squared plus b squared,” and it is astonishing that we can study geometry for a year without ever realizing that these “squares” refer, not to numbers multiplied by themselves, but to actual square areas. We are in this condition of cluelessness because, before we ever study geometry, we have already pre-interpreted it through the lens of the Cartesian grid and the operational equivalence of number and extension.

To lose (or be prevented from recognizing) the distinction between multitude and magnitude is to lose (or be prevented from attaining) a sense of the natural world as a world of distinct and integral living beings in their places; it is to be prepared to see it under the aspect of arbitrarily located quantities of material to be relocated according to our arbitrary will. For Aristotle, the formal integrity and unity of each living thing (which is to say its full flourishing as the kind of thing it is, adapted to the kind of place in which it lives) constitutes the good of that thing, toward which its development and actions and processes are ordered. In Platonic terms, one is the number of the good, and goodness is a constituent of the being of things. When Descartes reconfigures the natural world in terms of extension, and thus displaces natural unities in favor of quantitative relations, this amounts to evicting goodness as a principle of being. In other words, it is what I have elsewhere been describing as Gnosticism, and it prepares the way for Locke’s Gnostic economics, or the forgetting of limits and place.

The “argument” that supports such a view is embedded deep in our metaphysical imagination and habits of mind by a Cartesian mathematical education, one that perpetuates in our sensibilities Descartes’ own Gnostic alienation of spirit from world.

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Mark Shiffman was born in north Florida to the son of expatriated New York secular Jews and the daughter of small town, pillar of the community southern Presbyterians. After spending much of his childhood in Alaska and California, he discovered in his Tennessee adolescence, first reluctantly and then gratefully, that more than half his heart belonged to the South. He occasionally rediscovers this viscerally when his body descends below the Mason-Dixon line from his northern exile in Philadelphia, where he has also brought his wife into exile from her lifelong home of Chicago. They live in the Mount Airy section of Philadelphia with their two sons, having moved from one of the more successfully racially integrated neighborhoods in America (Hyde Park) to one of the most. Mark received his education from the McCallie School in Chattanooga and the surrounding mountains and trees, St. John’s College in Annapolis and the Santa Fe desert, Pendle Hill outside Philadelphia and the woods around Crum Creek, the University of Chicago and the icy prairie winds, and the Catholic Worker House and grimy streets of New York City. He is assistant professor in the Department of Humanities and Augustinian Traditions and affiliate faculty member in Classical Studies at Villanova University. He has also taught at Brooklyn College, Notre Dame, the University of Chicago and the University of Pennsylvania. His current projects include books on the political philosophy of Plutarch and on the meaning of modern individualism, as well as a translation of Aristotle’s On the Soul (Focus Press).

30 COMMENTS

  1. Mark,
    Do you have any idea just how long I’ll have to spend on this essay? This is excellent and deserves to be expanded and published for wide distribution.
    It appears the discovery and analysis of the gnostic distortion in existence is the first step to recover/recapture the truth of reality…and we have just begun!
    Voegelin was wrong about being right about gnosticism?

  2. Bob,

    The core of what I’ve said in this essay is more thoroughly worked out in Jacob Klein, Greek Mathematical Thought and the Origin of Algebra, and David Lachterman, The Ethics of Geometry. But I will at some point write more about the Gnostic angle.

    Speaking of which, could you expand on that last question about Voegelin?

  3. This is good!

    To put this more briefly, classical Aristotelian natural philosophy believes that only through formal and final cause (which are ultimately common, but viewed under the different aspects of “whatness” and “whyness”) can a thing be known. But modern natural philosophy, eschewing formal and final cause (in Newton’s Principia Mathematica there’s a Latin poem at the beginning decrying them as occult qualities, if I recall correctly) has instead to substitute quantifiable knowledge for this earlier, metaphysically-informed kind. If you can’t put number on it, it isn’t real: this is the creed of the modern world, of Descartes, and of Newton.

    This is also the root of our depersonalized and desacralized world which is at worst just “stuff” for our pillaging; at best the source of sentimental regard; but never numinous; never the basis for what is sacred.

    In order for modern viewpoint this to be true, time needs to be homogenized (there can be no difference in characteristics in “time” from one aeon to the next; it’s just temporal extension); and likewise space needs to share these same characteristics. For primitive peoples–and even for premodern ones like Aristotle to a lesser degree–this is not the case, or at very least much less the case. Time and space do not march endlessly, homogenously (which of course conveniently allows their graphing on the Cartesian coordinate system). Rather, they ebb and flow (as our subjectivity bears witness to on a daily basis).

    This essay makes important connections! Great job.

  4. In a nutshell, counter to abstracted Cartesian geometrical thinking that more narrowly defines data and breaks down the barriers between reality and arbitrary computation, man is, as the ancients’ believed, more than the sum of his parts and communities are more than their demographics.

    The inevitable counter-argument will be, but it works! We can predict physical realities with these Cartesian methods. We can increase our computational powers with high degrees of success, so how can it be bad? To which, I suppose the FPR response would be, that such powers, though useful, have led to things that are effective in the short term (industrial agriculture for instance, that produces great amounts of food) but is destructive to life and liberty in the long term (produces less nutritious food with the environmental consquences of soil erosion, pesticide and fertilizer runoff, and fossil fuel dependence).

    In other words, you are not saying that Cartesian geometry itself is evil, just dangerous when uninformed by a more “real” metaphysic?

    A wonderfully FPR essay (the first time FPR has been used as an adjective?). As PDGM said, important connections made!
    Thanks for making it through those big ol’ math books for those of us who are numbers challenged.

  5. This is certainly one of the best short essays on mathematics I have ever read. It comes down to the problem of the “one and the many” that so occupied Greek and Indian philosophy. The Hindus gave it a rather neat solution by making the many merely avatars of the one. That “solved” the problem, except that it isn’t a problem that ought to be solved; only contemplated.

    Mathematics deals with formal relationships, which are usually falsified materially. “1+1=2” is formally true, but “1 apple + 1 apple = 2 apples” is false, since unity implies identity, and no two apples are identical. The formalism is a short-hand way of saying “One thing with the quality of ‘apple-ness’ plus another thing with that same quality equals two things of the same quality,” but who wants to talk like that? The formalism is “true enough” for most purposes.

    Modern physics recovers ancient science, or has the possibility of doing so. Simone Weil thought that that Einstein’s relativity destroyed science, and so he did, but only the existing science. Neils Bohr and Werner Heisenburg recovered science. The “matter waves” turned out to be not material at all, but pure mathematical formulas that give the probability of finding anything in a particular place. But the interesting thing is that the thing isn’t there until it is found. That is to say, the particle, “particularity,” exists only in relationship. Until one wave, one set of probabilities, comes into relationship with another wave, no particle exists. The particle, prior to this, is neither here nor there; only when the wave meets another wave is the particle here and not there. Here, then, we have two things: a precise boundary between physics and metaphysics, and; an ontology where “being” is relationship. Hmm. What theology calls for this same ontology?

    It is silly to speak of the canonical variables (angular momentum and location) of a “single” particle, since such particles do not exist. Or if they do, we can’t know anything about them, since the only way to get any information is to send in another wave to locate the particle. Being is relationship, even at the physical level, a level that is never completely physical.

  6. Mark,

    My convuluted comment (the last sentence, above) refers to Voegelin’s classic work, New Science of Politics, published back in the early fifties where he pinpointed the modern deformation within Gnosticism where he says “…..the exploration of these problems was in the their beginning.” Well, I believe (and darn I can’t find the required Volume) he was attacked by critics and in time expanded his analysis to include not only gnosticism as the culprit but apocalypse, Neo-Platonism, and Hermeticism (tradition) as components of the distortion and derailment of modern man. He also cites a number of sources which, if you’d like to look them over, I can email to you. Also, my review of Professor Rossbach’s Gnostic Wars includes some pertinent material as well.
    I was never comfortable with EV’s abandonment of his original gnostic premise but, to be honest, it was just an inutition. How does one argue with a mind like that? But, this is why your approach is rather exciting and why, at least intuitively, I’m with you so far. Though one should take care with certain scholars who are derailed!
    This exploration of Gnosticism is important because EV never finished his work, he was relying on the upcoming generations and that’s you, dude!Talk about finding your niche!
    BTW, have you heard from D.W. Sabin? Man, I miss his acerbic comments, hope he’s ok?

  7. “Being is relationship, even at the physical level, a level that is never completely physical.”

    Right! It is the relationship between the genes that makes the cell, and it is the relationship between the cells that makes the organs, which make the organism (who is in fact a whole, because being is relationship).

    Have any of you read Christopher Alexander? In The Timeless Way of Building, he describes how the relationship between patterns of windows, doors, and walls creates a building. The relationship between patterns of buildings, streets, and people creates a place. From atom to cell to city, being is relationship.

    When we lose track of the necessity of larger and smaller patterns, we lose our sense of wholeness. If we forget the proper pattern of a city, we lose wholeness because of placelessness. If we corrupt the pattern of our cells, we lose wholeness to sickness. So Dawkins is wrong if he suggests that we are merely colonies of genes – we a colonies of genes, forming cells, forming organs, forming wholeness.

    Being is relationship, in organisms, communities, and cities. Let’s talk more about this.

  8. Bob: I have read little EV beyond New Science, but as you say, since this is starting to be my niche, I will have to press further on and in. Please do send me your Rossbach review. As for Sabin, since I know he has an axe to grind with algebra, I’ll have to try to lure him back to the porch.

    John and Patrick: Perhaps we need to start by clarifying the difference between a claim like “being is relationship” and a claim like “being is form.” For Aristotle, form is prior, and the parts (which are its material) have being because of their relationship to and within the form. But of course the forms of living things are, in the final analysis, their coordinated and sustained capacities for the life-activities that define them. These life-activities are what they are because of adaptation to where they are, but that includes adaptation to other living things where they are; so in a certain sense you might say being is relationship even for Aristotle. Plato, perhaps, is more explicit on this point. As I mentioned, in Platonic teaching, one is the number of the good; but two is the number of being, because everything that is a distinct something is so by being both the same with its own form and other than other forms, so that at least the relationship of otherness is intrinsic to its being.

    Thoughts?

  9. Not a mathemetician, me, rather a humble biochemist, but this paper from David Schindler at Villanova

    http://communio-icr.com/articles/PDF/DCS33-4.pdf

    captures a little of what was lost with enlightenment reformation of causality: evacuation of sense experience of meaning:

    “I intend to reflect on the transformation
    of the notion of causality in the seventeenth century and what this
    transformation implies for the significance of sense experience,
    which represents of course the foundation of the imagination. My
    thesis is that a mechanistic conception of the natural world evacuates
    sense experience of meaning, and therefore that the effort to
    cultivate the Christian imagination will be vain unless it is accompanied
    by a recovery of the ontological significance of goodness and
    beauty and thus by a critique of the popular view of the world
    inherited from classical physics.”

    and he lays the blame earlier than Descartes, at Galileo’s feet for the empirists coercion of praxis to deterministic theory:

    “Whereas, for Aristotle, motion is the
    actualization of a potential, and in this respect represents the
    unfolding of a nature, so that we have to describe it in the first
    instance as relative to that nature and thus in qualitative terms—e.g.,
    Aristotle demonstrates why circular motion is the most perfect and
    thus expected of the highest things—motion can have no intrinsic
    significance for Galileo: it is the homogenous monotony best
    described by number, successive units of the same.

    .
    and further

    “…what ’empirical’ means in the first and most fundamental sense is a cast of mind, a philosophical dispostion, before it deignates a real practice – but in point of fact this empirical method requires one to do violence to sense experience in a systematic fashion.”

    Similar critical thinking informed those of ther previous century like Menger in the Austrian school to reject a logical positivism in their theory of value. Unfortunately too many “establishment” conservatives reject their narrative out of hand as non-scientific, all economics must be deterministic. These reflections are key to grasp if the conservative movement is to rediscover the power and the beauty of human imagination, as a work of the Spirit inherent in each person endowed with an intellect and trained to use it appropriately, instead of being indoctrinated for serfdom (Hayek warned us, were we listening).

  10. “In classical Greek, in Aristotle’s physics, there is no word for ‘space’, only for ‘place.’ ”

    Yes. And the turn away from ‘place’ toward “space” is interestingly documented in Edward S. Casey’s tough, but rewarding, book _The Fate of Place_. For those of you inclined toward Continental Philosophy ( such as Heidegger, Merleau-Ponty) you might find his other works interesting as well.

    Great web-“place” by the way.

  11. Bob, I’m pleased there is a discussion of The Chairman on this site finally. Without reference to my library or notes, I would say that the development of EV’s thought led to a shift away from gnosticism as a complete explanatory device not because he believed he was wrong in his analysis, but because he came to reject his earlier methodology which interpreted history according to movements in “ideas.” Rather, he expressed a deeper philosophy of history in subsequent works encapsulized in the first sentence of his great work History & Order: “The order of history emerges from the history of order.”

  12. Clare: Of course Galileo is important in the way motion and nature get redefined mathematically. There are a number of key players in this story. But Descartes is, in a way, the most complete player. But I appreciate your use of Schindler as a corrective to my account, since that is how I have the good fortune to be able to use him often. I’ll have to mention that you invoked him when he’s over for dinner Monday.

  13. I would have benefited from a more detailed treatment of the classical idea of magnitude.

    “There is a marvelous beauty to the way the relations of figures are revealed when Euclid proves that the area of the square on the hypotenuse of the right triangle is the sum of the areas of the squares drawn on the other two legs. This beauty is entirely missing in the formula “c squared equals a squared plus b squared,””

    This reminds me of someone who only has a reading knowledge of a language. He understands a passage only by translating it into his native tongue rather than understanding it in its own terms and in its own “life.”

    Only in the case of math, our native tongue is Cartesian.

    Isn’t “translation error” a major distortion in modernity?

  14. Caleb, delightful and insightful comments on the beloved EV. He maintained the principle that ‘governed’ his work in the first three volumes of the seminal work, Order and History. But on Vol. four, The Ecumenic Age, a jaw-dropper as you know, he “revises his understanding of how the history of order must me analyzed and he altered his conception of history based on this revised analysis.” A note of interest: I’ve developed a fondness of FWC Schelling in reading a book (The Modern Philosophical Revolution)by a student of EV’s, Dr. David Walsh of CUA. EV would be proud of Walsh and his work is just breathtaking.
    In any event, I was reading in Vol. 33, EV’s discussion re: his writing of Order and History and he altered his thinking in writing this work, prior to the above. He was, as you know, not a rigid academic, his work always depended on the materials and on the truth. So Order and History is the product of a noetic epiphany brought about by EV’s reading of Schelling’s History of Myth. Here he abandoned the idea that ‘idea’ was anything but nonsense, rather there is only the history of experience-Schelling was THE father of Existentialim-and consequently EV’s work centered on the analysis of experience.
    I’m a contributor to Fritz Wagner’s Voegelin View website. Please check it out and I’m sure Fritz, who has amassed tons of EV material and makes it available to the faithful, would be delighted with your input, I know I would. Also, Fritz was a student of EV’s at ND! And, yes the exploration of EV’s thought is most promising, as is Drs. Shiffman and Peters analysis of the gnostic phenomenon, which is very important, seminial, work being published on your site!
    Best Wishes

  15. Bob, delighted to know you are hooked up with Fritz. I have know Fritz for many years and I believe I am supposed to be reviewing the Walsh book you reference for his new publication, though I haven’t actually received the volume yet. I was very disappointed by Walsh’s book Growth of the Liberal Soul, however, I hear tremendous things about his latest. -cs

  16. Mark, on the question of form, I have only questions, at least from the “physical” standpoint. Physics seems incapable of accounting even for physical forms, much less those “coordinated and sustained capacities for the life-activities.” We know a great deal of the chemistry of life, but cannot produce life from chemistry. But even for inanimate things, their form exceeds their chemistry. I think it possible that even form arises from relationship; what else could “coordinated capacities” mean? “Coordination” is a relationship. I am trying to keep relationship primary because I don’t want anything prior to it when speaking of the Trinity.

  17. Caleb, congrats on getting Walsh’s new one, I’ve read excepts and it’s very interesting. Somewhere along the line I’m going to have to beg it off the publisher. I’m glad to have your opinion of “The Growth of the Liberal Soul.” YOu might want to review that on FPR, I’d love to read it!
    I just got Gooch’s bio of the beloved Flannery for review.
    RCC

  18. Great comments, great piece. I’d just throw out that one of the very best books on this subject was never completed, but is still worth looking at: Husserl’s The Crisis of The European Science’s is both a very interesting development of his own phenomenology and a reversal of the modern understanding of mathematical meaning. It’s a tough read but its one of the most convincing refutations of the Cartesian-Kantian schema I’ve ever read. Its a real shame he never finished it. Sigh.

  19. I’ll second H.C.’s recommendation. You can start with the appendix, “The Origins of Geometry”, which briefly gives an idea of Husserl’s analysis of loss of meaning over time through what he calls “sedimentation”. This actually is a seminal text in the whole post-modern conversation (Derrida’s first book was a commentary on it), and it’s a good introduction to Jacob Klein’s very important book mentioned in my earlier comment. Husserl sees the symbols gradually and continually losing concrete reference to the original experiences that gave them meaning, whereas Klein points out a qualitative difference brought about by Viete, Stevin and Descartes in early modernity. Klein’s correction is a very important response to both Husserl and Heidegger, and is what initially convinced Leo Strauss of the radical rift between ancients and moderns.

  20. Good of John Medaille to ponder ‘relationship’ – may I offer by way of clarification: Square’s dream in Abbot’s Flatland,

    http://www.youtube.com/watch?v=INgA1uiYRS0

    where the King of Lineland knows no “relationship” of right or left, only forwards or backwards (aka that much vaunted “progress” we’re all supposed to be cheering for in Iran right? Or would that be left? Heck in Islam, does anyone care which way is up or down so long as its green, assertive, and raucously voiced?)

    An ethical voice (a moral if you will) presumes a point of departure which Mark criticizes Descartes for rendering as arbitrary, am I correct? The fault of liberal modernism is it assumes majority/superior quantity as a valid legitimator of “relationship” (or a function of agency, ie how many degrees of freedom an agent is permitted to exercise logically) Consider in chemistry the function of chirality, carbon’s 4-fold valency permits “quantities” of atoms to combine in right-handed or left-handed fashion on the tetrahedron, in forms healthy or lethal to nature, molecules that metabolize or metastasize, life-giving or toxic.

    Right relationship is key to our understanding of liberty in community. In economics a theory of value predicated on Cartesian mathematics (a ‘pris fixe’ definition of justice) will always be usurious, in that it usurps from the actor that degree of freedom to imagine the best (moral) use of the resources to be householded in the place and time of the circumstances he or she finds herself in, an obligation to sustain self (and ones dependents) in place that precedes any exchange of values to be anticipated from outcomes as yet indeterminate. Historicism collapses respect for the common good of a number of unique persons in magnitude of family bounded by a patrimony, into a concern for maximum good of the multitude of defined citizens of the state bounded by their gross national debt. The creative potential of the imagination instead of being the engine of human flourishing is denigrated to the rank of worker-bee/eunuch for the “soul of the hive” no?

  21. Cheeks,
    Though never fully “Well”, I am not unwell…… just off on a Y axis to yorn X axis for a spell. The youngest graduated from Univ. of Oregon and one cannot go all the way to Crunchy Eugene from Tight-arsed New England without both enjoying Portland’s restaurants or the remote beaches of Oregon where cell phones have yet to trill. The Geometry of the Willamette Pinot Noir Vineyards also much to commend them whilst the Concept and middle daughter are inside at a tasting rail pretending to understand as the vin master expounds upon that lovely concept of terroir. Posts 21′ on center, vines about 5′-6″ o.c and the rows around 5′ apart, all descending the hill rather than traversing it…..a scene with equal parts nature and mathematics conspiring to create an indisputable proof that the culture begins in a marriage of plant and the soil consecrated by the Farmer as Justice of the Peace.

    I am a non-imbibing Terroirista but fortunately, that little bit of fanaticism aint yet on the No Fly List.

    As to Algebra, my eternal gratitude is extended to the good Doktor Shiffman for finally clearing up for me why I could never fully trust algebra. And all these years I thought I was simply dense. There was always something a tad shifty to me about Algebra with its overtly hypothetical relations of x to y and its simultaneous dance with the finite and infinite. I felt more at home in geometry….always demanding the right to actually clutch or inhabit form with my mind to confirm it is really there. Talk about knee-jerk skepticism. Not seeing the real places of the little geometric boxes in algebra, I gave up on it early but know that without it, the map of our mind is less than complete.

    Patrick’s reference to Christopher Alexander is spot on. We respond well to a narrative architecture of compression and tensile expressions with relationships between windows and doors based upon human scale because we are conversant with the fundamentals of this architectural dialogue. It is not alien. Approach the modern deracinated and omni-place strip mall and the scene is devoid of dialogue and human scale because it holds no relationship beyond the merely transactional. It’s scale is with the automobile and speed. The breadth of a stride is irrelevant because the turning radius of a delivery truck is all that really matters.

    Had I been aware that mathematics can be a social science, I might have tried harder at penetrating my barriers against the arid axis of algebra.

    Lastly, with this site’s steerage of my reading list toward Augustine and Voegelin , you have bucketed me but good damn ye.

  22. D.W.’s back, all is well with my….!

    Boy, from now on you let us know! This is a bunch of crap, you coulda been wrapped around a telephone pole on a deserted stretch of highway just a-moulderin’!

    I thought someone aggravated you, but on second thought I knew that was unlikely…you being the aggravator!

    Shiffman’s going all the way with gnosticism, this is cutting edge, new materials, the analysis….somewhere in the bosom of Eternal Being Voegelin smiles! By the time we’re done, we’ll need a good priest and a rabbi to get you right with the Lord!

    My best to the wife, shalom, and congrats on the youngest’s graduation!

  23. Just a word about Voegelin and gnosticism. Voegelin was interested in the symbol of The Beginning. Not just as the first cause, and certainly not just as a beginning in time, but as a symbol for a kind of starting point of existence, which is why in his later work he focused on an analysis of the structure of consciousness. Gnosticism therefore is an example of derailment or deformation of consciousness, but just looking at examples of deformation is not a sufficient way of looking at and understanding the structure of consciousness. As a practical matter, problems such as gnosticism are the starting point of philosophy. Voegelin philosophized because he was reacting to the political situation, the fact that people and governments would kill masses of people, particularly their own people, for no good reason. But why? It is not just a problem with motivation, or even a problem of orientation, although this is getting closer.

    So in his last published volume (unfinished) he works on the question of the structure of consciousness, and there is a significant emphasis on the problem of the in-between nature of reality, working off of the beginning phrase of Genesis 1, “In the beginning…

    So while there is always an astounding treatment of individual thinkers and historical “movements” in Voegelin, and a wealth of historiographic materials, the essential Voegelin, shall we say, is concerned about the structure of the in-between nature of consciousness and of reality as a whole, and what is it about the structure of consciousness that pushes beyond its structure.

    So he never, in his later work, asserts that all of these cultural problems we are experiencing can be traced back to one thinker, or one historical period. That would be a gnostic interpretation of history. But it is a common problem among (we) conservatives, looking for some period of history, or one thinker that caused everything to go wrong with the idea that we can somehow get behind or go before that thinker and make everything right again.

  24. I would prefer it if rather than looking to Euclid’s method of deductive proof for geometry as an alternative to Descartes and to move into anti-Euclidean geometry of Gauss and Riemann.

    However, since the main subject here is Descartes I don’t think any discussion of Descartes would be complete without talking about his main detractor Leibniz.

    When scientist stop trying make the constructs of Descartes and Newton congruent with the discoveries of Einstein and Planck and instead just accept the fact that Leibniz was right all along perhaps we have a bit progress and the so-called Enlightenment’s hostility toward Christianity will pass from our culture and we will get Western civilization back.

  25. Just in case anyone else comes across this post, I’ll respond to O Watson:

    Not sure what you mean by “real mathematics”, but if 4 years studying history of mathematics from primary sources in college and 4 years as a teaching assistant in mathematics courses at the University of Chicago counts, then I suppose I must have picked up some knowledge of it along the way.

    Also not sure what you are trying to say with your “arrows” metaphor. Geometry developed for a couple thousand years without algebra, and algebra for a few hundred without being applied to geometry. Some people in the 16th and 17th centuries decided that the two could serve as representational equivalents, and I have raised the question whether important things are lost in the results of that decision.

    Now I would ask whether you have made the intellectual experiment, as I have, of thinking through ancient geometry (Euclid and Apollonius in particular) with your thoughts as insulated as possible from the conceptual filter of Cartesian algebraic geometry. That would be a first step to what I would call “understanding real mathematics,” and it would be great if they taught that in graduate math programs.

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