Breaking Mathematical Sense

I asked him to outline the algo [algorithm] for me,” one junior accountant remarked about her derivatives-trading Porsche driving superior, “and he couldn’t, he just took it on faith.” “Most kids have computer skills in their genes … but just up to a point … when you try to show them how to generate the numbers they see on screen, they get impatient, they just want the numbers and leave where these came from to the main-frame.

Arvidsson, Adam. The Ethical Economy (p. 3). Columbia University Press. Kindle Edition.


Mathematicians, as far as I can see, are not terribly interested in the philosophy of mathematics. They often have philosophical views, but they are usually not very keen on challenging or developing them—they don’t usually consider this as worthy of too much effort. They’re also very suspicious of philosophers. Indeed, mathematicians know better than anyone else what it is that they’re doing. The idea of having a philosopher lecture them about it feels kind of silly, or even intrusive. (Roi 2017, 3)

So we turn to people who have something to do with mathematics in their professional or daily lives, but are not focused on mathematics. Such people often have some sort of vague, sometimes naïve, conceptions of mathematics. One of the most striking manifestations of these folk views is the following: If I say something philosophical that people don’t understand, the default assumption is that I use big pretentious words to cover small ideas. If I say something mathematical that people don’t understand, the default assumption is that I’m saying something so smart and deep that they just can’t get it. (Roi 2017, 3-4)

There’s an overwhelming respect for mathematics in academia and wider circles. So much so that bad, trivial, and pointless forms of mathematization are often mistaken for important achievements in the social sciences, and sometimes in the humanities as well. It is often assumed that all ambiguities in our vague verbal communication disappear once we switch to mathematics, which is supposed to be purely univocal and absolutely true. But a mirror image of this approach is also common. According to this view, mathematics is a purely mechanical, inhuman, and irrelevantly abstract form of knowledge. (Roi 2017, 4)

I believe that the philosophy of mathematics should try to confront such naïve views. To do that, one doesn’t need to reconstruct a rational scheme underlying the way we speak of mathematics, but rather paint a richer picture of mathematics, which tries to affirm, rather than dispel, its ambiguities, humanity, and historicity. (Roi 2017, 4)

(….) The uncritical idolizing of mathematics as the best model of knowledge, just like the opposite trend of disparaging mathematics as mindless drudgery, are both detrimental to the organization and evaluation of contemporary academic knowledge. Instead, mathematics should be appreciated and judged as one among many practices of shaping knowledge. (Roi 2017, 4-5)

Some Ideas on Education in the Management Sciences, Management Science, 17: b2-4.

A Vignette: Option Pricing and the Black-Sholes Formula

Be a market maker—try to buy and sell very quickly, and take benefits from the spread between the bid and offer.

— Senior Morgan Stanley Trader cited in Nicholas Dunbar‘s The Devil’s Derivatives.

The point of the following vignette is to give a concrete example of how mathematics relates to its wider scientific and practical context. It will show that mathematics has force, and that its force applies even when actual mathematical claims to not quite work as descriptions of reality…. The context of this vignette is option pricing. An “option” is the right (but not the obligation) to make a certain transaction at a certain cost at a certain time. For example, I could own the option to buy 100 British pounds for 150 US dollars three months from today. If I own the option, and three months from today 100 are worth more than 150 dollars, I will most probably simply discard it. Such options could be used as insurance. The preceding option, for example, would insure me against a drop in the dollar-pound exchange rate, if I needed such insurance. It could also serve as a simple bet for financial gamblers. But what price should one put on this kind of insurance or bet? There are two narratives to answer this question. The first says that until 1973, no one really knew how to price such options, and prices were determined by supply, demand, and guesswork. More precisely, there existed some reasoned means to price options, but they all involved putting a price on the risk one was willing to take, which is a rather subjective issue. (Roi 2017, 6)

In two papers published in 1973, Fischer Black and Myron Sholes, followed by Robert Merton, came up with a reasoned formula for pricing options that did not require putting a price on risk. This feat was deemed so important that in 1997 Scholes and Merton were awarded the Nobel Prize in economics [see The Nobel Factor] for their formula (Black had died two years earlier). Indeed, “Black, Merton and Scholes thus laid the foundation for the rapid growth of markets for derivatives in the last ten years”—at least according to the Royal Swedish Academy press release (1997). (Roi 2017, 6-7)

But there’s another way to tell the story. This other way claims that options go back as far as antiquity, and option pricing has been studied as early as the seventeenth century. Option pricing formulas were established well before Black and Scholes, and so were various means to factor out putting a price on risk (based on something called put-call parity rather than the Nobel-winning method of dynamic hedging, but we can’t go into details here). Moreover, according to this narrative, the Black-Sholes formula simply doesn’t work and isn’t used (Derman and Taleb 2005; Haug and Taleb 2011).

If we wanted to strike a compromise between the two narratives, we could say that the Black-Scholes model was a new and original addition to existing models and that it works under suitable ideal conditions, which are not always approximated by reality. But let’s try to be more specific. (Roi 2017, 7)

The idea behind the Black-Scholes model is to reconstruct the option by a dynamic process of buying and selling the underlying assets (in our preceding example, pounds and dollars). It provides an initial cost and a recipe that tells you how to continuously buy and sell these dollars and pounds as their exchange rate fluctuates over time in order to guarantee that by the time of the transactions, that money one has accumulated together with the 150 dollars dictated by the option would be enough to buy 100 pounds. This recipe depends on some clever, deep, and elegant mathematics. (Roi 2017, 7)

This recipe is also risk free and will necessarily work, provided some conditions hold. These conditions include, among others, the capacity to always instantaneously buy and sell as many pounds/dollars as I want and a specific probabilistic model for the behavior of the exchange rate (Brownian motion with a fixed and known future volatility, where volatility is a measure of the fluctuations of the exchange rate). (Roi 2017, 7)

The preceding two conditions do not hold in reality. First, buying and selling is never really unlimited and instantaneous. Second, exchange rates do not adhere precisely to the specific probabilistic model. But if we can buy and sell fast enough, and the Brownian model is a good enough approximation, the pricing formula should work well enough. Unfortunately, prices sometimes follow other probabilistic models (with some infinite moments), where the Black and Scholes formula may fail to be even approximately true. The latter flaw is sometimes cited as an explanation for some of the recent market crashes—but this is a highly debated interpretation. (Roi 2017, 7-8)

Another problem is that the future volatility (a measure of cost fluctuations from now until the option expires) of whatever the option buys and sells has to be known for the model to work. One could rely on past volatility, but when comparing actual option prices and the Black-Sholes formula, this doesn’t quite work. The volatility rate that is required to fit the Black-Sholes formula to actual market option pricing is not simply past volatility. (Roi 2017, 8)

In fact, if one compares actual option prices to the Black-Sholes formula, and tries to calculate the volatility that would make them fit, it turns out that there’s no single volatility for a given commodity at a given time. The cost of wilder options (for selling or buying at a price far removed from the present price) reflects higher volatility than the more tame options. So something is clearly empirically wrong with the Black-Sholes model, which assumes a fixed (rather than a stochastic) future volatility for whatever the option deals with, regardless of the terms of the option. (Roi 2017, 8)

So the Black-Sholes formula is nice in theory, but needn’t work in practice. Haug and Taleb (2011) even argue that practitioners simply don’t use it, and have simpler practical alternatives. They go as far as to say that the Black-Sholes formula is like “scientists lecturing birds on how to fly, and taking credit for their subsequent performance—except that here it would be lecturing them the wrong way” (101, n. 13). So why did the formula deserve a Nobel prize? (Roi 2017, 8)

Looking at some informal exchanges between practitioners, one can find some interesting answers. The discussion I quote from the online forum Quora was headed by the question “Is the Black-Sholes Formula Just Plain Wrong?” (2014). All practitioners agree that the formula is not used as such. Many of them don’t quite see it as an approximation either. But this does not mean they think it is useless. One practitioner (John Hwang) writes:

Where Black-Sholes really shines, however, is as a common language between options traders. It’s the oldest, simplest, and the most intuitive option pricing model around. Every option trader understands it, and it is easy to calculate, so it makes sense to communicate implied volatility [the volatility that would make the formula fit the actual price] in terms of Black-Sholes…. As proof, the exchanges disseminate [Black-Sholes] implied volatility in addition to data.

Another practitioner (Rohit Gupta) adds that this “is done because traders have better intuition in terms of volatilities instead of quoting various prices.” In the same vein, yet another practitioner (Joseph Wang) added:

One other way of looking at this is that Black-Sholes provides something of a baseline that lets you compare the real world to a nonexistent ideal world…. Since we don’t live in an ideal world, the numbers are different, but the Black-Sholes framework tells us *how different* the real world is from the idealized world.

So the model earned its renown by providing a common language that practitioners understand well, and allowing them to understand actual contingent circumstances in relation to a sturdy ideal. (Roi 2017, 9)

Now recall that practitioners extrapolate the implied volatility by comparing the Black-Sholes formula to actual prices, rather than plug a given volatility into the formula to get a price. This may sound like data fitting. Indeed, one practitioner (Ron Ginn) states that “if the common denominator of the crowd’s opinion is more or less Black-Sholes … smells like a self fulfilling prophecy could materialize,” or, put in a more elaborate manner (Luca Parlamento):

I just want to add that CBOE [Chicago Board Options Exchange] in early ’70 was looking to market a new product: something called “options.” Their issue was that how you can market something that no one evaluate? You can’t! You need a model that helps people exchange stuff, turn[s] out that the BS formula … did the job. You have a way to make people easily agree on prices, create a liquid market and … “why not” generate commissions.

The tone here is more sinister: the formula is useful because it’s there, because it’s a reference point that allows a market to grow around it. (Roi 2017, 9)

But why did this specific formula attract the market, and become a common reference point, possibly even a self-fulfilling prophecy? Why not any of the other older or contemporary pricing practices, which are no worse? Why was this specific pricing model deemed Nobel worthy? (Roi 29017, 10)

The answer, I believe, lies in the mathematics. The formula depends on a sound and elegant argument. The mathematics it uses is sophisticated, and enjoys a record of good service in physics, which imparts a halo of scientific prestige. Moreover, it is expressed in the language of an expressive mathematical domain that makes sense to practitioners (and, of course, it also came at the right time).

This is the force of mathematics. It’s a language that the practitioners of the relevant niches understand and value. It feels well founded and at least ideally true. If it is sophisticated and comes with a good track record in other scientific contexts, it is assumed to be deep and somehow true. All this helps build rich practical networks around mathematical ideas, even when these ideas do not reflect empirical reality very well. (Roi 29017, 10)

(….) [I]f we want to understand the surprising force of mathematics demonstrated in this vignette, we need to engage in a more careful analysis of mathematical practice. (Roi 29017, 10)

24 thoughts on “Breaking Mathematical Sense

  1. Dave Marsay

    ” If I say something philosophical that people don’t understand, the default assumption is that I use big pretentious words to cover small ideas. If I say something mathematical that people don’t understand, the default assumption is that I’m saying something so smart and deep that they just can’t get it.”

    “I believe that the philosophy of mathematics should try to confront such naïve views. To do that, one doesn’t need to reconstruct a rational scheme underlying the way we speak of mathematics, but rather paint a richer picture of mathematics, which tries to affirm, rather than dispel, its ambiguities, humanity, and historicity. ”

    As an old-fashioned mathematician, I agree that there is a real problem here. But I think we need to distinguish between mathematics per se and what Keynes called ‘pseudo-mathematics’. The philosophy of mathematics as such seems harmless, and the comments of philosphers generally misguided or pointless. But beware pseudo-maths!

    There used to be a view that the real world must somehow conform to Euclidean Geometry, but these days mathematics as such makes no claims at all about the real world. All it can do is to expose the logical consequences of whatever claims domain experts make. To correct this in the sciences, one needs a philosphy of science. More generally, one needs a philosophy such as that of Russell ( ) or Whitehead ( ). It is for philosophers, not mathematicians, to critique such works. For myself, they seem adequately to confront some naive views that Roi finds problematic.

    So we need – and perhaps already have – mathematically credible philosophy, rather than philosphy ‘of’ mathematics. (At least, that’s how I see it.)

    1. Meta Capitalism

      I appreciate the links Dave, will give them a careful look. I enjoyed reading Whitehead long ago and have notes somewhere stored away. And yes, I have seen a lot of philosophy that makes me wonder, whatever do they mean? I was careful when introducing my daughters to philosophy to choose carefully. One of my favorite were from Cathcart and Klein, such as the GOP motto of unprincipled conservatism perfectly summed up by Groucho Marx,

      “These are my principles; if you don’t like them, I have others.”

      ~ Cathcart & Klein. Plato and a Platypus Walk Into a Bar: Understanding Philosophy Through Jokes (Kindle Locations 22-23). ABRAMS. Kindle Edition.

    2. Meta Capitalism

      I don’t know if you will see this or not Dave, but I will try/reply anyway. I think your distinction regarding discerning the difference between mathematics per se and “pseudo-mathematics” (I understand more fully what you mean now) as described by Keynes. I don’t take such a dismal view of philosophy, but take the point one must sift the chaff from the wheat carefully. But I am more interested in a comment you made on RWER in your dialogue with Ken Zimmerman:

      I quite accept that as a body of work mathematics is a human product that needs to understood in its proper context, including culture. But:
      1. Is there not ‘in effect’ and in some obscure sense a possible reality behind the various versions of mathematics that might be thought of, at least as a working hypothesis, as ‘real mathematics’ in much the same way that some people think that there is some reality that has some influence on physics beyond the thoughts of physicists?
      2. Based on my own experience, it seems to me that that the ‘cultural distance’ between British, European, American, Chinese and Russian mathematicians, when talking about mathematics, is insignificant compared with that between mathematicians and social scientists. Do you have any views on this?

      The reality behind culture is in my view the reality of mind and its emergent ability on the human level to develop language. This innate ability transcends any specific culture. But this enters the domain of philosophy which extends beyond the limits of science and even enters into the domain of living spiritual experience. Several posts on this site such (Penrose 1990; Stapp 2007, Stapp 2009).

      1. Dave Marsay

        Got it! Will look at your refs,, especially Penrose. (i have a Penrose count of 2, by some measure.)

        I certainly hope that there is something that can get us out of the dismal straight-jacket that our cultures seem to put us in!

      2. Dave Marsay

        On Penrose, I quite agree that a humans can do what no Turing machine, algorithm or ‘definite method’ can. But don’t see why some more general machine might not be able to solve some mathematical problems that a Turing machine can’t, So I agree with your paragraph without comprehending Penrose at all (and I’ve tried).

        I’ve scanned the first of your Stapp quotes. In so far as he is critical of classical and modern Physics andan advocate of post-modernism, it is perhaps worth noting that Keynes ( ) came to a similar view after talking to the same group of Physicists, leading to a reform of mathematics. Confusingly, though, what he calls ‘modern mathematics’ is what Stapp would call ‘post-modern’. I haven’t read enough of him to know if he gets this. (As Ken notes, many see mathematics through a modernist ‘cultural lens’, which is unfortunate.)

        From a modern mathematical perspective, Stapp uses terms such as ‘beliefs’, ‘observers’ and ‘knowledge’ in ways that I am unable to make much sense of. But you avoid such ambiguous (possibly non-logical and meaningless) terms. I’m never quite sure if such words are used because he thinks we think we understand them, or because he thinks he does despite Russell and Whitehead, or if he is simply innocent of logic. What do you think?

      3. Meta Capitalism

        There are more things in heaven and earth, Horatio,
        Than are dreamt of in your philosophy.
        – Hamlet (1.5.167-8), Hamlet to Horatio

        To the unbelieving materialist, man is simply an evolutionary accident. His hopes of survival are strung on a figment of mortal imagination; his fears, loves, longings, and beliefs are but the reaction of the incidental juxtaposition of certain lifeless atoms of matter. No display of energy nor expression of trust can carry him beyond the grave. The devotional labors and inspirational genius of the best of men are doomed to be extinguished by death, the long and lonely night of eternal oblivion and soul extinction. Nameless despair is man’s only reward for living and toiling under the temporal sun of mortal existence. Each day of life slowly and surely tightens the grasp of a pitiless doom which a hostile and relentless universe of matter has decreed shall be the crowning insult to everything in human desire which is beautiful, noble, lofty, and good. (Urantia Book 102:0.1, paraphrasing Bertrand Russell’s “A Free Man’s Worship”)

        But such is not man’s end and eternal destiny; such a vision is but the cry of despair uttered by some wandering soul who has become lost in spiritual darkness, and who bravely struggles on in the face of the mechanistic sophistries of a material philosophy, blinded by the confusion and distortion of a complex learning. And all this doom of darkness and all this destiny of despair are forever dispelled by one brave stretch of faith on the part of the most humble and unlearned of God’s children on earth. (Urantia Book 102:0.2)

        This saving faith has its birth in the human heart when the moral consciousness of man realizes that human values may be translated in mortal experience from the material to the spiritual, from the human to the divine, from time to eternity. (Urantia Book 102:0.3)

        Perhaps Dave your trouble with terms like “belief,” “observers,” and “knowledge” are more a matter of your limitations and need to conform everything to your understanding of human nature from a materialist philosophical perspective and the need to fit it into the box of logic and mathematics that is the problem? Creative insight into universe mind-meanings and mind perceived spiritual-values transcends both logic and mathematics, but neither is illogical or inconsistent with the logic of reason or the findings of true science. There is a huge domain of world class studies in the fields of the humanities, philosophy, comparative religion, even scientific fields as exhibited by the Vienna Series in Theoretical Biology and so on that sheds light on such questions beyond mere logic and mathematics. Perhaps you have limited your intellectual range according to Russel and Whitehead? Russel and Whitehead are not the totality of the universe when it comes to the philosophy of science.

        Mathematics is the language of science, which is founded upon logic and reason. But mind has capabilities that transcend the mere logic of mathematics and can creatively gain insights that while perfectly reasonable are not ultimately founded upon the logic of mathematics, to wit Computability: Turing, Gödel, Church, and Beyond.

        Here is the dilemma of materialism. Faced with the very real emergent reality of genuinely creative thought that transcends mere algorithmic logic materialists themselves are driven to their own philosophical “beliefs” into their own faith-based-belief in the “mystic philosophies that mind is emergent from the brain and that mind, brain and body constitute the individual existing in a monist reality (Markey-Towler 2018, 10).” Such was the source cited by Shiozawa and used as the foundation of his new work Microfoundations of Evolutionary Economics.

        The observation of emergent properties is descriptive; it tells us little factually of the origin or cause of emergent reality. To say that mind “emerged” from matter explains nothing. If the universe is merely a mechanism and mind is unapart from matter, we would never have two differing interpretations of any observed phenomenon. The concepts of truth, beauty, and goodness are not inherent in either physics or chemistry. A machine cannot know, much less know truth, hunger for righteousness, and cherish goodness.

        Nobel laureate Sir Peter Medawar said, “Catastrophe apart, I believe it to be science’s greatest glory that there is no limit upon the power of science to answer questions of the kind science can answer.” (P. B. Medawar, The Limits of Science, p. 87)

        Science may be physical, but the mind of the truth-discerning scientist is at once supermaterial. Matter knows not truth, neither can it love mercy nor delight in goodness (spiritual values and mind meanings). Moral convictions based on philosophical thinking and spiritual enlightenment rooted in living human experience are just as real and certain as mathematical deductions based on physical observations, but on another and higher level. If men were only machines, they would react more or less uniformly to a material universe. Individuality, much less personality, would be nonexistent.

        And therein lies the limitations of a mechanistic materialist philosophy; its inability to reconginze the limits of science and the limits of its language of mathematics grounded in logic and reason. It is not unreasonable to recognize the very real human values of truth, beauty, and goodness, for we strive to bring these realities to our loved one’s, our children and and our spouses, everyday. The crisis the world is facing today is not one of science or even economics let alone the narrow question of the methodology of economics. Rather we are facing a crisis of moral, ethical, and spiritual values and insight into how we as a species are going to transition from one level of civilization to another without destroying ourselves in the process of this never ending evolution of biological things, intellectual meanings, and spiritual values.

        We hold to the following philosophical notions in this work. The mind is that element of our being which experiences our place in the world and relation to it. We are conscious when we are aware of our place in and relation to the world. We hold to a mix of the “weak Artificial Intelligence” and mystic philosophies that mind is emergent from the brain and that mind, brain and body constitute the individual existing in a monist reality. (Markey-Towler 2018, 10)

        If one is going to take consciousness seriously then one is faced with the question of “belief,” mind-meanings and mind-values that exist and transcend the logic of mathematics. You yourself have many such “beliefs” which puzzles me how you can claim to not understand them other than to think you are doing your philosophy unconsciously.

        If consciousness is accepted as real, it seems reasonable that one would allow for an active consciousness, for us to be aware of the experience of thinking and to engage in that experience. If we didn’t allow for engaged and active thought in consciousness, then consciousness would seem to be a passive “ghost in the machine” sort of consciousness. (Markey-Towler 2018, 8)

        There is no universal science of man. Science is a never ending quest like the asymptote coming ever closer but never reaching its limit. Philosophy is the mediator between science and religion — living personal insight into those values worth pursuing (I call these spiritual values, spiritual insights, or wisdom if you like).

        The ability of mathematics to transcend culture is rooted in the cosmic reality of mind and the fact that one of the emergent properties of mind on the human level is its abililty — its reality sensitivity — to perceive (1) Causation: the reality domain of the physical senses, the scientific realms of logical uniformity, the differentiation of the factual and the nonfactual. This is the mathematical form of the cosmic discrimination. (And don’t let Ken’s cultural relativism confuse you in its verbosity; even primitive human minds were sensitive to and sought causation, but without science could only confuse and conflate the material and non-material levels of reality giving rise to superstition, magical thinking, and primitive religion.) (2) Duty—the reality domain of morals in the philosophic realm, the arena of reason, the recognition of relative right and wrong. This is the judicial form of the cosmic discrimination. Adam Smith’s more important work was his The Theory of Moral Sentiments. It is a poverty of ideals (moral, ethical, and spiritual) not ideas that are threatening the world today. (3) Worship—the spiritual domain of the reality of religious experience, the personal realization and recognition of spirit values, the assurance of eternal survival, the ascent from the status of servants of God to the joy and liberty of the sons of God. It is this level (and one should look at the best, not the worst, of the worlds saints and religious teachers in the world’s religions) that leads one human to love and serve another beyond their tribal loyalties.

        The Golden Rule is found throughout all the world’s religions and philosophies to one degree or another. Humanism too is a religion, if one defines religion as that which one holds of supreme value. Therefore, it is possible for humanists, religionists, spiritualists, etc. to join together in collectively determining what values and meanings are going to guide science (and economics) as we face our current civilizational crisis that far transcends the narrow debates transpiring on RWER, in my view. That is the kind of knowledge and wisdom I am pursuing, and economics is just one small corner in this quest.

  2. Meta Capitalism

    I think you may have misread Stapp (2007) on post-modernism. Unless of course I have misread him! 😉 My understanding is that he is critiquing post-modernism as having gone to far in one direction, and that more recent findings in QM have corrected somewhat this swing towards the kind of “relativism” that has been popularized and was not the real meaning of QM. He writes:

    This turning away by science itself from the tenets of the objective materialist philosophy gave impetus to, and lent support to, post-modernism. That view, which emerged during the second half of the twentieth century, promulgated, in essence, the idea that all ‘truths’ were relative to one’s point of view, and were mere artifacts of some particular social group’s struggle for power over competing groups. Thus each social movement was entitled to its own ‘truth’, which was viewed simply as a socially created pawn in the power game. (Stapp 2007, 141-142)
    The connection of post-modern thought to science is that both Copenhagen quantum theory and relativity theory had retreated from the idea of observer-independent objective truth. Science in the first quarter of the twentieth century had not only eliminated materialism as a possible foundation for objective truth, but seemed to have discredited the very idea of objective truth in science. But if the community of scientists has renounced the idea of objective truth in favor of the pragmatic idea that ‘what is true for us is what works for us’, then every group becomes licensed to do the same, and the hope evaporates that science might provide objective criteria for resolving contentious social issues. (Stapp 2007, 142)
    This philosophical shift has had profound social and intellectual ramifications. But the physicists who initiated this mischief were generally too interested in practical developments in their own field to get involved in these philosophical issues. Thus they failed to broadcast an important fact: already by mid-century, a further development in physics had occurred that provides an effective antidote to both the ‘materialism’ of the modern era, and the ‘relativism’ and ‘social constructionism’ of the post-modern period. In particular, John von Neumann developed, during the early thirties, a form of quantum theory that brought the physical and mental aspects of nature back together as two aspects of a rationally coherent whole. This theory was elevated, during the forties — by the work of Tomonaga and Schwinger — to a form compatible with the physical requirements of the theory of relativity. (Stapp 2007, 142)

    He goes on to present and new view which he relates came after the Copenhagen interpretation of QM that gives a more real (e.g., the “efficacious conscious human choices”) role to subjective human experience.

    A common core feature of the orthodox (Copenhagen and von Neumann) quantum theory is the incorporation of efficacious conscious human choices into the structure of basic physical theory. How this is done, and how the conception of the human person is thereby radically altered, has been spelled out in lay terms in this book, and is something every well informed person who values the findings of science ought to know about. The conception of self is the basis of values and thence of behavior, and it controls the entire fabric of one’s life. It is irrational, from a scientific perspective, to cling today to false and inadequate adequate nineteenth century concepts about your basic nature, while ignoring the profound impact upon these concepts of the twentieth century revolution in science. (Stapp 2007, 143)

    Human personality is real; it is a reality in the universe. Mind is real; it is a reality in the universe. A purely mechanistic philosophy of life and the universe cannot be scientific because science recognizes and deals only with materials and facts. Philosophy is inevitably super-scientific. Man is a material fact of nature, but his life is a phenomenon which transcends the material levels of nature in that it exhibits the control attributes of mind and the creative qualities of spiritual values. It is the fact of mind-meanings and human personality and its associated spiritual-values that interjects into material reality spontaneity and radical uncertainty.
    As Stapp notes, it is irrational to propose a theory of everything that leaves out the most important thing we know in the universe (e.g., ourselves and our loved ones), which is our conception ourselves, others, and society and the fact of the reality of the “efficacious conscious human choices” we make as individuals and societies that by-and-large determine the kind of world we shall live in.
    Any scientific interpretation of the material universe is valueless unless it provides due recognition for the scientist. No appreciation of art is genuine unless it accords recognition to the artist. No evaluation of morals is worth while unless it includes the moralist. No recognition of philosophy is edifying if it ignores the philosopher, and religion cannot exist without the real experience of the religionist.
    The scientist, not science, perceives the reality of an evolving and advancing universe of energy and matter. This is essentially, as far as I can tell in my honest and sincere attempt to understand Stapp, what he is saying.

  3. Dave Marsay

    Rob, I may well have misread Stapp. For example:
    “This turning away by science itself from the tenets of the objective materialist philosophy gave impetus to, and lent support to, post-modernism. That view, which emerged during the second half of the twentieth century, promulgated, in essence, the idea that all ‘truths’ were relative to one’s point of view, and were mere artifacts of some particular social group’s struggle for power over competing groups. Thus each social movement was entitled to its own ‘truth’, which was viewed simply as a socially created pawn in the power game.”

    My experience of people who call themselves post-modernists is that they are even more evangelical that most evangelical Christians. They tend to be very dismissive and unrespectful of my ideas, I seems to me that they talk ‘at’ me rather than with me. (Although I admit to being diifficult to understand.)

    You say “Perhaps Dave your trouble with terms like “belief,” “observers,” and “knowledge” are more a matter of your limitations and need to conform everything to your understanding of human nature from a materialist philosophical perspective and the need to fit it into the box of logic and mathematics that is the problem? ” One possible interpretation of this is that you think I have a ‘materialist philosophical perspective.” Do you? If so, why? (I’ve not read any further., for reasons which may be obvious.But maybe you’d like to critique my blog, at


    1. Meta Capitalism

      My apologies if I assumed too much. I am not a Christian nor a member of any particular religious tradition. I have read the Urantia Book from a very young age (sixteen) and studied the history and philosophy of religion and comparative religion for many years. I almost pursued it as field of professional study, but opted for business instead in the end for pragmatic reasons. Then around the 1990s I turned to the study of the history and philosophy of science, particularly the history of biology and evolutionary theory with a focus on evolutionary developmental biology and epigenetics. Only recently (three years ago) have I turned to studying economics.

      I have found your blog most interesting. You are thoughtful on a deep level. I actually want to spend more time there reading when I can find the time, and I have an interest in pursuing mathematics on a deeper level. I don’t post because I feel I owe your posts more time, a more thoughtful reply, and until I find the time to provide a carefully crafted reply I thought best to just read. I think mathematics is not merely a cultural creation used to control things. Sure, there is a pragmatic side of mathematics, but it is more than just a pragmatic tool. I have examples in mind in the history of science and particle physics that I think do show it does create cross-cultural bridges and does transcend mere cultural relativism. Unfortunately, I would have to retell the story from memory as my core science library remains in the states while I am in Japan.

      To be honest, from the limited readings on your blog I did not think you were a mechanistic materialist philosophically speaking, nor a post-modernist, or social constructivist. One can give proper place to material reality without being a philosophical materialist; one can give proper place to relative cultural concept frames and the influence of social context without becoming a post-modernist or social constructivist. I find some of these totalizing worldviews just as limiting and having blinders as religious fanaticism and theological dogmatism. I found you deeply thoughtful (seeking understanding, truth) and touching on core questions. Perhaps I have misread your statement (or read to much into it) about “belief,” “observers,” and “knowledge” when your characterize them as “non-logical and meaningless terms.” Of course, in philosophy, the first lesson is to define one’s terms. Is that what you mean? In our daily lives we use many terms that are not carefully defined, but does that really make them meaningless and non-logical?

      No doubt words can be ambivalent, mean different things to different people, and scholars, philosophers, etc., have different nuanced definitions depending on one’s field or specialty. Take the terms faith and belief. These are often equated in the popular imagination and even some dictionary definitions. But look little deeper in the field of comparative religion and one finds nuance and the classical work of Wilfred Cantwell Smith who studied the historical usage of the terms faith and belief throughout human history and wrote his classical book “Faith and Belief” on the history of these two terms.

      I think Stapp uses the term “observer” to mean a human personality, an embodied personal human being of self-conscious mind and reflective capability participating in a material world manipulating the instruments used to run quantum experiments.

      A rough definition of belief, in my experience, defines belief as the intellectual assent to a propositional statement that may or may not be true. Knowledge is a bit trickier of course, because it is more than mere belief in that it is supposed to accord with reality and there is a general assumption one’s knowledge is true. Of course, the older and wiser one becomes and the more history of science one studies the more one realizes how provisional, how imperfect, and how tentative knowledge really is to one degree or another.

      1. Dave Marsay

        Rob, Glad to see I haven’t been totally misunderstood.

        “Of course, in philosophy, the first lesson is to define one’s terms. Is that what you mean?” Kind of. It is “of course” impossible to define one’s terms in the sense in which many people seem to think one can. But one should do one’s best and always try to be aware of any residual ambiguity. Hard!

        This is one aspect of ‘radical uncertainty’. I rather hope that a social sciences or economics 101 that avoids the need to study logic or philosophy or anthropology to an advanced level first could be devised. But I do think that all economists should at least be aware of these issues, so that if they do happen to become significant we don’t all suffer the consequences.

        Incidentally, I don’t mind honest but ‘mistaken’ comments on my blog: It helps me to understand where others are coming from better, and gives me an opportunity to improve my texts.;-)

      2. Meta Capitalism

        What timing! I was just on your blog reading your posts on pragmatism. It’s funny, it seems the more we define terms sometimes the more questions we raise, the more nuanced the definitions become and the more residual ambiguity we find 😉 But one side of me says, “Hey, don’t overcomplicate things. Complicate them pragmatically!”

        The more economics history I read the more original ambiguity I am finding leaving only the residual ambiguity of whether or not this field of so-called knowledge can really play a positive role in the current crisis the world is facing. I think we need to step back and ask some fundamental questions again, like, What is economics? What is its purpose? Is it a philosophy (a bad one at that) pretending to be science? What pragmatic useful contributions has it made both past and present? How might it make pragmatic useful contributions moving forward?

        It is not all bad though. I have found some encouraging works such as Kate Raworth’s Doughnut Economics. There are others I hope to comment upon later. Perhaps some Amazon reviews or something.

      3. Meta Capitalism

        You note it is,

        impossible to define one’s terms in the sense in which many people seem to think one can. But one should do one’s best and always try to be aware of any residual ambiguity.

        I think your posts on how to develop a “fruitful form of pragmatism” (here, here, here, and here) lead to one being able to define terms “good enough” with the caveat they are always open to modification based upon new information. I will take the time to read all of them more carefully and given them some thought and comment on your blog. Thanks for posting them Dave.

      4. Dave Marsay

        Rob, leaving aside the contents of economics for a moment, I guess much of what I’m trying to matters to economics is that what previous generations thought was ‘good enough’ may now be a part of the problem. I’m also suggesting some alternatives (mostly what Keynes et al suggested) that may be good enough for now, or at least ‘as good as it gets’, but one would really need to engage with our current issues to make a judgement.

        Thanks for engaging.

      5. Meta Capitalism

        Here is where I think mathematicians like yourself (and Lars) are helping. This is my understanding up to this point with my limited knowledge. Please help me where I am in error or lacking insight. Including providing links to your blog where I should read. Lars has with lazer focus exposed the logical and mathematical assumptions underlying mainstream economics that are in error. This is not a critique of mathematics per se; I totally agree with your idea of pseudo-mathematics (now that I understand) vs. the proper use of mathematics. I have read his book (have you? I only ask to see if I can skip lengthy quotes and use shorthand when referencing it) not just once but several times. I think I understand it now.

        I see a critical role for mathematicians like yourself and Lars at a minimum helping the public not be fooled by misleading pseudo-mathematics masquerading as science. Whether there can ever be a mathematics of radical uncertainty I don’t know. It seems that was the role of statistics. Here is my question to you. Do you see any limits to reason, logic, and mathematics in the human domain? For example, can mathematics deal with human greed, fraud, and power and corruption in the business and political realms, and if so how so? What role does mathematics play in the building of moral and ethical character and the discernment and pursuit of the human values that would make our civilization better? As someone who has studied at great length business on many levels, read extensively in the business literature, mathematics has had little to add to the moral and ethical dimensions that play such a large role in the history of classical political philosophy and early thinking of the founders of economics. Adam Smith’s more important work was after all his Theory of Moral Sentiments. The recent failures of capitalism (e.g., the GFC) were not so much mathematical or logical failures as failures of institutional integrity and outright unethical greed, corruption, and fraud. This is well documented. Where does this fit into your role?

        When I say “good enough” I don’t mean blindly accepting mainstream economic theory or for that matter anything. I mean when discussing complex subjects, after defining one’s terms, there is a point at which the clarity is good enough for some level of discussion. Some argue (and I won’t name who, you know) mathematics is just a social construct for controlling things. It provides no real insight into reality. I am not of that mind. I feel very differently. While I recognize mathematics is a language that humans invent as needed, that does not mean I cannot see the fact that the logic and ability to creatively invent such mathematics that have again and again provided insights into the nature of material reality, allowed us to make predictions long before they were empirically confirmed, also tells us something about not only mathematics, but the creative mind and the universe itself. I will leave it there for now, and see if you have ideas that might shed light on this thought stream.

      6. Dave Marsay

        Rob, Be aware that Lars and I are coming from very different places. I once asked him to ‘peer’ review a paper of mine, which he was kind enough to ‘pass’. But it seems to me I am not in his target audience, so revealing our differences to the world may not be a good idea: Instead I am trying to articulate things which I hope Lars will ‘let go’. to identify some sort of consensus. (As with others!) In doing so I am allowing myself plenty of wriggle room so that I can change my position, if need be, without anyone noticing.(Seems wise.)

        I speculate that whatever Lars and I do agree on might be ‘good enough’ to provide a basis for the reform of economic methodology, but we do need to make it accessible to the wider audience, somehow. (Not my specialist area, sorry.)

        I haven’t made sense of any economist other than Keynes, and in doing so I seem to ended up somewhere quite different from most ‘Keynesians’. So quoting other people doesn’t help that much, if at all. But you seem to be making sense. (Maybe you could try to make some sense of ‘you know who’? Maybe he’s making an honest effort, but handicapped by his ‘culture’? 😉 )

        P.S. I do have some more ideas, but need to refine their presentation. I wouldn’t want anyone to get the wrong idea.

      7. Meta Capitalism

        Reductionism has three types; the first two are philosophic, and the third is a common scientific strategy. Ontological reductionism denies the being of higher levels of reality: “So-called spiritual reality is nothing but a psychological experience; psychological experience is nothing but a biological process; and a biological process is nothing but a set of biochemical events.” Epistemological reductionism claims to explain a higher-level science wholly in terms of a lower-level science, for example, “Scientific theories using psychological terms can, in principle, be reduced to scientific theories using only biological terms; in the future scientists will be able to explain biological terms solely in terms of chemistry and physics.” So-called “non-reductive materialism” accepts ontological reductionism but rejects epistemological reductionism. Methodological reductionism says, “In order to have a coherent and rigorous science, we exclude any hypotheses about spirit. For the purposes of this research program, we restrict our conclusions about spiritual experience to the language of neuroscience.” Note that religionists and atheists can in good faith co-author reports of methodologically restricted research. Nevertheless, methodological reductionism may be problematic, too, because it raises a crucial question in the philosophy of science. Science, Scientia in Latin, means knowledge, which implies knowledge of the region of reality that it addresses. It would be absurd to claim to plumb the meaning of human action by the methods of chemistry. Should not the method appropriate to a given region of reality be attuned to that region itself? This observation does not imply that chemistry says nothing important about action, but it does imply that the meaningfulness of what chemistry tells us depends on a prior understanding of action itself. Making a commitment to a scientific method on account of its quantitative precision or other epistemological advantages can hobble access to the region to be known. (Jeff Wattles, personal communication, 2/14/2013, emphasis added.)

        Lars has been fairly open on his philosophical views on his personal blog. He calls himself a Critical Realist and has posted links to Roy Bhaskar, which I assume, he agrees with his philosophically, more or less. Essentially it is a kind of “non-reductive materialism” [Mind is an emergent property of matter].
        The concept of agency itself is very important to critical realists and they see mind as an emergent power of matter and reasons as causally efficacious in producing actions. (Roy Bhaskar, Critical Realism). All social events occur simultaneously on for plains. These are the plains (i.e., four plains of social being):

        1. Material transaction with nature.
        2. Social interaction between people.
        3. Social structure.
        4. Stratification of the embodied personality.

        Roy uses terms like MetaReality, DemiReality (world of illusion and oppression), Unrecognized realm of trust and solidarity. The World of DemiReality is tacitly underpinned by a world of trust and solidarity. The world of ‘trust and solidarity’ which we can identify in the world of here and now and form the basis of a society of universal human flourishing. The Idea of seriousness; unity of theory and practice. A philosophy we can act on.

        I am a critical realist but of a very different philosophy 😉

      8. Dave Marsay

        It seems to me that there is a great deal of sense in what you, Lars and Ken are saying. I have a huge amount of experience at getting people to doubt those they had trusted, and if I can’t get them to doubt themselves I tell myself this is not my fault: they are just too pig-headed. But what’s the point?

        It would be useful if I had some way of telling which person or culture was most promising to engage with, but I don’t find social scientists (broadly defined) very helpful. But they do say things like ‘proximity promotes propinquity’.

        Does your philosophy have anything to say that might help me? If so, it would seem like a good deal for me if you only wanted help with your maths homework. 😉 😉

      9. Meta Capitalism

        In the mid-eighteenth century, when the Enlightenment had all but won the day, when German philosopher Georg Hegel’s dialectic helped philosophers make tidy sense of contradictory quandaries, and when the Evangelical Lutheran Church of Denmark celebrated christenings with cocktails, a gawky misfit and satiric genius walked the streets of Copenhagen. Søren Kierkegaard took on the ecclesiastical pomposity, bourgeois fatness, and spiritual lethargy of his day. He served not only as a spy for God, but as a moral assassin. He shot his darts at smug, complacent churchgoers as fat geese that had forgotten how to fly. In contrast to a public that wanted no disturbance to their lives, his pseudonymous Johannes Climacus suggested, writing “pen in cheek,” that what “the world, confused simply by too much knowledge, needs is a Socrates.” Irony may lead to true edification; religious satire to genuine reform. (Lindvall, Terry. God Mocks: A History of Religious Satire from the Hebrew Prophets to Stephen Colbert (Kindle Locations 231-238). NYU Press. Kindle Edition.)

        Don’t despair Dave, you have made me doubt myself on more than one occasion and no doubt will do so again 😉 I love your quip, all the erudite social science enlightening us with “proximity promotes propinquity,” as it reminds of the tautology so frequently used in the just-so stories of Ultra-Darwinists, the pretzel logic of the Ultra-Darwinists in their attempts to get around the tautological definitions of natural selection like “survival of those that survive.” (Robert G. B. Reid. Biological Emergences: Evolution by Natural Experiment (Vienna Series in Theoretical Biology) (p. 58). Kindle Edition.)

        If you are looking for certainty in the realm of knowledge then might I recommend the Church of Sisyphus 😉 When a certain series of discoveries emerged in the 1990s in the field of biology I realized it was about to undergo a major scientific revolution so I spent tens of thousands of dollars building a professional library (mostly of graduate level books) and devoted decades studying the science, history, and philosophy of biology with a passion that infused my daughters such they both went into biology (oldest graduating in neurobiology and youngest pursing same in Canada). I learned much and there are some things I can say with certainty. One of those things is that natural selection is not the cause of novelty in evolution, that it only functions upon existing variation, but is not the cause of that variation. For me the survival of the fittest is interesting and natural selection (the sieve of already existing phenotypic variation) is real, but it is only a proximate cause, the more interesting question being the arrival of the fittest or the real source of variation or the ultimate cause of hereditary variation that lies in a deeper understanding of homology, epigenetics, developmental biology (evo-devo) and the emerging science(s) of comparative genomics and epigenetic evolution. It is a history I hope to write more about in the near future. Yet I also have many questions remaining, many doubts, and don’t expect answers to them soon, if ever. I also learned though that scientific revolutions are not really so much revolutions as slow moving funerals. So, in the end, despite knowing more and gaining certainty about my understanding of the cause(s) of evolution I still have doubts and uncertainties. Such is the nature of human knowledge and always will be; relative degrees of certainty and uncertainty, knowingness and doubt. Welcome to the Church of Sisyphus 😉

        Personally, I will finish up reading the corpus of the WEA library and stop engaging on RWER as I have plumbed its depths to the degree that I am now finding myself covering the same tedious territory in the discussions. Ken’s cultural relativism and social constructivism is at its core self-contradictory and leads to nothing and nowhere. It offers no way out and I personally sense that is the reason of the underlying vitriol and violent tendencies in his rhetoric. Ken is shallow thinker in my view; I am not fooled by his verbosity. He disdains philosophy (he said to openly) and heaps contempt upon religion (he is of the Dawkins/Hitchens ilk) and therefore he unconsciously is philosophically inconsistent.

        The irony of his mechanistic philosophy of life and the universe is that it cannot be scientific because science recognizes and deals only with materials and facts. Philosophy is inevitably super-scientific. Man is a material fact of nature, but his life is a phenomenon which transcends the material levels of nature in that it exhibits the control attributes of mind and the creative qualities of spirit.

        Bluntly put Dave I don’t think it is going to far to say mind transcends matter; especially given that is exactly what Stapp is saying and that quantum mechanics teaches us the simple fact that the consious observer, the human personlity interacting with material reality through the medium of mind-choice, must by the very nature of reality be part of the so-called equation 😉 That is the essence of what Stapp (and many others) are telling us.

        Either one takes this insight seriously and then begins the process of adjusting one’s philosophy accordingly or one retreats to some form of materialism or pantheism or physical monism, etc. For me personally mechanistic materialism was not an option due to experiential and philsophical reasons.

        Conceptualizing Cells

        We should all take seriously an assessment of biology made by the physicist David Bohm over 30 years ago (and universally ignored):

        “It does seem odd … that just when physics is … moving away from mechanism, biology and psychology are moving closer to it. If the trend continues … scientists will be regarding living and intelligent beings as mechanical, while they suppose that inanimate matter is to complex and subtle to fit into the limited categories of mechanism.” [D. Bohm, “Some Remarks on the Notion of Order,” in C. H. Waddington, ed., Towards a Theoretical Biology: 2 Sketches. (Edinburgh: Edinburgh Press 1969), p. 18-40.]

        The organism is not a machine! Machines are not made of parts that continually turn over and renew; the cell is. A machine is stable because its parts are strongly built and function reliably. The cell is stable for an entirely different reason: It is homeostatic. Perturbed, the cell automatically seeks to reconstitute its inherent pattern. Homeostasis and homeorhesis are basic to all living things, but not machines.

        If not a machine, then what is the cell?

        (Woese, Carl R., Author. Evolving Biological Organization. In Microbial Phylogeny and Evolution: Concepts and Controversies. (Jan Sapp, ed.). Oxford: Oxford University Press; 2005: 100.)

        If our ever-enlarging scientific and biological knowledge is telling us simple cells are not machines then why believe that human mind is a mere algorithmic machine, and that humans are merely complex machines?

        Ken is a materialist who preaches that the human imagination (the source of creativity in human culture) is ultimate and absolute and therefore the totality of reality is viewed through this distorting lens of social constructivism. But whence comes such human imagination without mind? Either mind and our free-will experience are delusions and mere artifacts of matter or they are real. And if they are real then our philosophy must imagine more than mere mechanistic materialism.

        But in my view there does exist in the universe a reality the transcends cultural relativism and social constructivism and its understanding lies in the realms of philosophy and religion. But that is another story:

        Mortal man is a machine, a living mechanism; his roots are truly in the physical world of energy. Many human reactions are mechanical in nature; much of life is machinelike. But man, a mechanism, is much more than a machine; he is mind endowed and spirit indwelt; and though he can never throughout his material life escape the chemical and electrical mechanics of his existence, he can increasingly learn how to subordinate this physical-life machine to the directive wisdom of experience by the process of consecrating the human mind to the execution of the spiritual urges of the indwelling Thought Adjuster [indwelling divine spirit]. (The Urantia Book 118:8.2)

        My tentative conclusion after studying the economics over the last three years is that the crisis we are facing today is a crisis of civilization and is much bigger than the mere quibbles over methodology in sociology. This is not a scientific crisis; we already have the requisite scientific knowledge to solve our problems many times over. It is rather, a crisis of moral, ethical, and spiritual values and meanings. And the solution to these questions are found, in my view, in the fields of history of ideas, philosophy of science, ethics, and religion, and finally, and most importantly, in a new religious philosophy of living that leads human beings to seek through close intimate contact to learn to love and serve one another despite our differences intellectually, socially, and culturally.

      10. Dave Marsay

        Kierkegaard and Bohm have often seemed to provide assistant to those who want to reach across cultures. As a mathematician I struck by Wikipedia’ take on SK: “Logic and human reasoning are inadequate to comprehend truth”. Fighting talk!

        Actually, though, Logic has come on along way since his time. Wikiepedia also presents him as commending faith to supply what logic can not. I dont’ have a particular problem with. It’s just that one needs the right logic, and hence the gap may no longer be quite as large as some people seem to think (not just Ken!) So, what would SK and DB make of logic and maths as they are now? (And which logic and maths?)

        Ken would perhaps say that SK and DB are helpful (subject to the above quibble) because of the influence they have had on the culktures of those they seem to help. (And it does seem to me you and I have a lot in common.) John Lennon (my new favourite philosopher 😉 ) suggests that the way to resolve this is to to reach out to others of different cultures (not just Japanese!) Good idea?

        (Also, I would suggest you look at my blog and read whichever of my sources are most accessible for you for yourself, before considering my comments – just in case Ken has a point.)

      11. Meta Capitalism

        Thanks Dave, I do need to take some time reading your blog. I have several threads I have plans on reading while we ride out this super typhoon!

        I am not much of a fan of Wikipedia. Logic and mathematics certainly do tell us something about truth, at least in my understanding they do. That is why I think I can learn something valuable reading your blog and sharing. I gave several books on mathematics I brought to Japan and hope to get to soon.

        You are right I think about cross-cultural sharing. Yes, it is a good idea. I lived in South Korea for five years, Japan for as many, Taiwan, the Nederland, and of course the US. I think the more we get to know other cultures and languages, the more we see we share common human values and traits despite our differences.

        I don’t mean to be dismissive of Ken, for he does have some cogent insights. I just grow bored with his book length soap box posts. And after a while it becomes rather more of the same over and over.

        Well, time to batten down the house and get ready for the kami kazi 🙂

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  4. Meta Capitalism

    Very interesting. I think we can help each other Dave, perhaps you helping me more than I you 😉 But we need a more discrete place I think. If you contact me at my private junk email robreno at hotmail dot com I will send my real private email where we can share ideas and thoughts and insights more discretely.

    A long while back on RWER one of the now-and-then posters made a statement that has stuck with me and which I agreed with. He said something to the effect that the best we can do is make our philosophical assumptions explicit for they largely determine the conceptual options open to us in our choices of methodology in tackling scientific questions. No doubt that is a very loose interpretation of his comment and meaning. But it is what I took away from it.

    This accords with my life experience. I have found when discussing subjects like science, philosophy, and religion determining the other’s basic philosophical orientation helps me understand why they argue (in the best sense of its meaning) the way they do. In many cases those philosophical assumptions pigeonhole one’s mind into a rather narrow set of conceptual options. We are all subject to such bias in our thinking (this is where logic and mathematics are supremely serviceable in exposing such bias).

    I have always used mathematics as a tool; learning whatever mathematics was needed to solve a particular problem at hand. But now as I grow older I want to approach it from a philosophical perspective and see if this allows me to better appreciate its beauty, power, and nature. Perhaps you can help in this regard. Brought a few books with me to Japan to do just that.

    I could, after exchanging a private key, setup a private place on my blog were only you and I could see the posts. Just a thought.


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