set theoretical foundationalism

Steve Awodey proposes category-theory based structuralism as a substitute for the set-theory based foundationalism:

We can say that the "foundational perspective," to which we are proposing an alternative, is based on the idea of building up specific "mathematical objects" within a particular "foundational system," in such a way that:

1. There are enough such objects to represent the various kinds of numbers as well as the spaces, groups, manifolds etc. of everyday mathematics, and

2. There are enough laws, rules, and axioms to warrant all of the usual inferences and arguments made in mathematics about these things, as well as at least some of the most obvious "rounding off" statements dealing with features of the system itself (like the well-foundedness of all sets, a question that does not arise in non-set-theoretic mathematics).

As opposed to this one-universe, "global foundational" view, the "categorical structural" one we advocated is based instead on the idea of specifying, for a given theorem or theory only the required or relevant degree of information or structure, the essential features of a given situation, for the purpose at hand, without assuming some ultimate knowledge, specification, or determination of the "objects" involved. The laws, rules, and axioms involved in a particular piece of reasoning, or a field of mathematics, may vary from one to the next, or even from one mathematician or epoch to another. The statement of the inferential machinery involved thus becomes a (tacit) part of the mathematics; functional analysis makes heavy use of abstract functions and the axiom of choice, some theorems in algebra rely on the continuum hypothesis many arguments in homology theory are purely algebraic, once given the non-algebraic objects that they deal with; theorems in constructive analysis avoid impredicative constructions; 19th century analysis employed other methods than modern-day analysis, and so on. The methods of reasoning involved in different parts of mathematics are not "global" and uniform across fields or even between different theorems, but are themselves "local" or relative.

Thus according to our view, there is neither a once-and-for-all universe of all mathematical objects, nor a once-and-for-all system of all mathematical inferences. Are there, then, various and changing universes and systems? How are they determined, and how are they related? Here I would rather say that there are no such universes or systems; or rather, that the question itself is still based on a "foundationalist" preconception about the nature of mathematical statements...

Theorems state connections, relations, and properties of the structures involved: group, topological, continuous actions, etc. The proof of a theorem involves the structures mentioned, and perhaps many others along the way, together with some general principles of reasoning like those collected up in logic, set theory, category theory, etc. But it does not involve the specific nature of the structures, or their components, in an absolute sense. That is, there is a certain degree of "analysis" or specificity required of the proof and beyond that, it doesn't matter what the structures are supposed to be or to "consist of"- the elements of the group, points of the space, are simply undetermined.

This lack of specificity or determination is not an accidental feature of mathematics, to be described as universal quantification over all particular instances in a specific foundational system as the foundationalist would have it - a contrived and fantastic interpretation of actual mathematical practice (even more so of historical mathematics!). Rather it is characteristic of mathematical statements that the particular nature of the entities involved plays no role, but rather their relations, operations, etc. - the "structures" that they bear - are related, connected, and described in the statements and proofs of theorems. It is a theorem in topology that the first homology group of an arcwise-connected space is naturally isomorphic to the abelianization of the fundamental group of the space. This statement doesn't depend on the specific points of the space, or even on the specific space; it is about a connection between homology and homotopy. In this sense, mathematical statements (theorems, proof, etc. even definitions) are about connections, operations, relations, properties of connections, operations on relations, connections between relations on properties, and so on.

Set theory provides a universal setting for all of mathematics at two great expenses.

1) It strips off the commonalities across the different structures. From a set theoretical perspective, the only thing that rings and abelian groups have in common is the fact that they are sets. Knowing that two things are made of elements brings zero insight regarding their structural similarities. From a categorical perspective, a single functor (namely the forgetful functor) from the category of rings to the category of abelian groups highlights all the structural similarities present. (Note that a ring is an abelian group with some further structure.)

Category theory allows you to work on structures without the need first to pulverise them into set theoretic dust. To give an example from the eld of architecture, when studying Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don't do is begin by imagining it reduced to a pile of mineral fragments.

Corfield - Towards a Philosophy of Real Mathematics (Page 239)

2) For the sake of founding things on simpler things, it explains the clear with the obscure.

In a definite sense, all mathematics can be derived from axiomatic set theory... This view leaves unexplained why, of all the possible consequences of set theory, we select only those which happen to be our mathematics today, and why certain mathematical concepts are more interesting than others. It does not help to give us an intuitive grasp of mathematics such as that possessed by a powerful mathematician. By burying, e.g. the individuality of natural numbers, it seeks to explain the more basic and the clearer by the more obscure.

- Hao Wang ass quoted in Tool and Object (Page 26)

In order to see how contrived it is to define natural numbers in set theory, let me explain to you how it is actually done!

You first need to lay down the axioms of Zermelo-Fraenkel set theory. ZFC is just a list of infinitely many statements written in the language of first order logic. One of these statements is the Axiom of Infinity which guarantees the existence of a set with the following two properties:

- It contains the empty set.

- Whenever it contains an ordinal it also contains the successor of that ordinal.

An ordinal is a tricky notion to grasp. The important point to know is that ordinals are well-ordered and each ordinal consists solely of all the ordinals that precede it. Hence given an ordinal A you can legitimately talk about the successor ordinal, namely the ordinal coming right after A. But the fact that every ordinal has a successor does not imply that every ordinal is a successor. It may be impossible to pinpoint the ordinal which comes right before A. (In such a case A is called a limit ordinal.)

Define a natural number M as an ordinal satisfying the following property: If an ordinal A precedes M or is equal to M (in other words if A is contained in M), then A is either the empty set or a successor ordinal. It is easy to see that each natural number M is contained in the set spawned into existence by the Axiom of Infinity. So the collection of all natural numbers resides in a set. Hence, using Axiom of Comprehension, we can carve out that collection and consider it as set on its own. There you have the set of natural numbers N!

Of course you may still wonder whether N really captures the intuitive idea of what natural numbers are. The answer is easily seen to be affirmative if you are willing to assume that the intuitive idea is completely characterized by the Peano axioms (which by the way can only be expressed in second-order logic).

This set theoretical construction involves a truly bottom-up approach. It also happens to be truly inexplicable to non-mathematicians!

In category theory, the set of natural numbers is not built from ground-up. It is treated as a single impenetrable object and the characterization is entirely external. (The key is to capture the recursive character of the set of natural numbers without invoking the numbers themselves.) This approach has an important added advantage: We do not have to restrict ourselves to the category of sets. We can define a natural number object in any category that has a terminal object.

Let ε be an arbitrary category with a terminal object 1. (Note that the terminal object in Set is simply the one element set {*}.) The natural number object is defined as a triple (N,z:1→N,s:N→N) where the object N and morphisms z,s satisfy the following universal property. (Think of z as the function that send * to number zero, and s as the successor function sending number M to M+1.) For any triple (A,q:1→A,f:A→A), there exists a unique morphism u:N→A such that the following diagram commutes:

If ε was the category Set, then u would be the function taking number M to f(u(M-1)). There is only one such function since we need to have u(1)=f(u(0))=f(u(z(*)))=f(q(*)). (For more details, read this expository paper by Mazur.)

non-associative operators

Non-associative binary operators can encode more information than associative ones. (Same relationship holds between non-commutative and commutative binary operators too. Both facts stem from the meta principle that introduction of symmetries causes loss of information.) For instance, Henry M. Sheffer provided an alternative, more economic axiomatization of Boolean algebras using his stroke operator |:

This non-associative operator encodes the usual Boolean algebra operators as follows:

Note that a more economic axiomatization may not necessarily be more practical for the working mathematicians. For instance, proving certain identities may now take longer. On the other hand, showing that some structure qualifies to be Boolean algebra may now be easier.

why wittgenstein

Why have I been picking on Wittgenstein lately? Is it because he was an arrogant fellow with no sense of empathy? True, I hate such people. But there seems to be something else behind my hostility.

Wittgenstein and I have strangely many things in common. I realized this only last year when I read his biography written by Ray Monk.

- He was passionate and sincere. He hated pretentiousness.
- He preferred an informal and succinct writing style.
- He avoided joining any groups and despised elite circles.
- He was never satisfied with anything he produced.
- He was generally pessimistic and did not like paternal authorities.
- He had an authoritative father who built a huge industrial fortune.
- He was greatly influenced by the suicide of a friend.
- He had an hernia operation while young.
- He had immense respect for people doing practical things.
- He encouraged his friends to pursue non-academic careers.
- He had trouble controlling his temper during discussions.
- He enjoyed interrupting speakers and pointing out their mistakes.
- He had a peculiar ability to see old things in a new light.
- He had a strong mental imagery ability.
- He worked on the foundations although he knew it was futile.

Moreover he was a self-loathing person. To a certain extent this is also true of me. The reason why I feel hostile to Wittgenstein may be due to this fact. I want to pick on him, because we have so many things in common.

philosophical impairments

First a long quote (I have omitted all the in-prose references to increase readability):

In obvious reference to Frege's Grundgesetze der Arithmetik, Wittgenstein pointed out, in conversation with Schlick and Waismann, that there is a third possibility, alongside the views of the formalists and Frege:

For Frege the alternative was this: either we deal with strokes of ink on the paper or these strokes of ink are signs of something and their meaning is what they go proxy for. The game of chess itself shows that these alternatives are wrongly conceived - although it is not the wooden chessmen we are dealing with, these figures don't go proxy for anything, they have no meaning in Frege's sense. There is still a third possibility, the signs can be used the way they are in a game.

The accusation is certainly unfair, since Frege was aware of the analogy with the game of chess; it was made by Thomae in a passage which Frege himself quotes and discusses at length. This passage is worth putting side by side with Wittgenstein's remark:

For the formalist, arithmetic is a game with signs, which are called empty. That means they have no other content (in the calculating game) than they are assigned by their behaviour with respect to certain rules of combination (rules of the game). The chess player makes similar use of his pieces; he assigns them certain properties determining their behaviour in the game, and the pieces are only external signs of this behaviour.

According to Thomae, signs in arithmetic derive their meaning from the rules of arithmetic. This sounds strikingly similar to Wittgenstein's position as states above, but there could not be an idea more alien to Frege than this. For Frege the situation ought to be exactly the reverse: rules are derived from meanings. As he construed the formalist position, rules are simply created, stipulated arbitrarily; they have no basis: "We do not derive these rules from the meaning of the signs, but lay them down on our own authority, retaining full freedom and acknowledging no necessity to justify the rules". Frege even accuses Thomae of pre-supposing meanings in his formal arithmetic:

Although numerical signs designate something, this can be ignored, according to Thomae, and we can regard them simply as pieces manipulated in accordance with rules. If their meaning were to be considered, this would supply the grounds for the rules; but this occurs behind the scenes, so to speak, for on the stage of formal arithmetic nothing of the sort can be seen.

Wittgenstein was not in any strict sense a formalist. For example, one aspect of the formalist doctrine which has no equivalent in his writings is the insistence on the tangibility of signs: "Is Mathematics about signs on paper? No more than chess is about wooden pieces". But it is no exaggeration to say that the whole of Wittgenstein's later philosophy of language goes against the Bedeutungskörper conception of meaning upon which Frege's "arithmetic with content" is based. His approach is antithetical to Frege's: "Rules do not follow from an act of comprehension". When speaking about comprehension, Frege's usual turn of phrase is that we "grasp" (Fassen, Erfassen) meanings, but he never explained what he meant by that. In Philosophical Grammar, Wittgenstein pointed out this lacuna:

In attacking the formalist conception of arithmetic, Frege says more or less this: these petty explanations of the signs are idle once we understand the signs. Understanding would be something like seeing a picture from which all the rules followed, or a picture that makes them all clear. But Frege does not seem to see that such a picture would itself be another sign, or a calculus to explain the written one to us.

Marion - Wittgenstein, Finitism, and the Foundations of Mathematics (Pages 177-179)

Clearly, Wittgenstein did not know how to take things easy. When Frege says he just grabs meanings, he probably did not lie. In fact, Frege was one of those few intellectuals who was brutally honest with himself. Here is an illustrative anecdote by Russell who discovered the paradox that undermined the foundations of Frege's arithmetic:

As I think about acts of integrity and grace, I realise there is nothing in my knowledge to compare with Frege's dedication to truth. His entire life's work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his fundamental assumption was in error, he responded with intellectual pleasure clearly submerging any feelings of personal disappointment. It was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known.

Without any doubt, Wittgenstein had an immense power of mental imagery. Therefore understanding had a pictorial nature for him. On the other hand, Frege had a lesser imaging ability. Hence understanding had a mysterious quality for him.

...for Frege any complete account of how expressions can possess meaning at all would have to involve a complete account of how we can grasp and understand pure thoughts – and it does not appear that he thought this question could be sufficiently answered with recourse to language simply because for Frege language itself presupposes certain rational capacities in order to be capable of expressing thoughts. This seems to be precisely why Frege repeatedly takes refuge in various metaphors to indicate the existence of certain rational faculties that enable us to grasp a thought, like the mysterious “power of thinking” that he talks about in “Thought”. Indeed, in a draft dating from 1897 he explicitly describes the act of thinking as “perhaps the most mysterious of all”, and adds that he regards the question of how it is possible as “still far from being grasped in all its difficulty”. (Source)

David Berman has a wonderful paper that illustrates how some of the irreconcilable disagreements between philosophers may have been due to actual cognitive differences. Apparently the fact that imaging capability varies widely across the population was known in the scientific literature for more than 150 years... I had no idea about it! Here are some extracts from this paper for those who are too lazy to read it in its entirety. While reading these pieces, keep in mind that Berman has absolutely zero imaging ability.

(Note that I have omitted all the in-prose references.)

We can picture imaging ability (and images themselves) as on a scale of 0 to 10... Those with 10, the maximum, have photographic and eidetic imagery. Although most photographic imagers are also eidetic, the two kinds of images and imaging ability needn’t go together. What photographic imagery is should be reasonably clear. Briefly, an image is eidetic if it has more rather than less independence from the mind or will of the imager, enabling him to scan or move around his images. And while the images of most eidetics are of photographic or near photographic detail, they needn’t be. Probably the most famous case of someone with both photographic and eidetic ability was the subject of A. R. Luria’s classic study, the Mind of a Mnemonist, who could form images that were indiscernible from objects seen in the physical world, both in their detail and substantiality. Moving to the minimum extreme of the spectrum, to 0, there are those individuals who have no images.

In his own classic work, The Principles of Psychology (1890), William James writes:

Until very recent years it was supposed by all philosophers that there was a typical human mind which all individual minds were like, and that propositions of universal validity could be laid down about such faculties as ‘the Imagination’. Lately, however, a mass of revelations have poured in, which make us see how false a view this is. There are imaginations, not ‘the Imagination’, and they must be studied in detail.

And although it was Gustav Fechner who was the ‘first-breaker of ground in this direction’, it was the publication of Galton’s work, according to James, that ‘may be said to have made an era in descriptive Psychology’. James then quotes at length Galton’s description of how he came to make his crucial discoveries concerning imagery.

To make a long story short, Galton devised a two-page questionnaire which he distributed widely, not only in England but also in France, America and elsewhere. The aim of Galton’s questions was to determine the vividness, detail, location, etc. of mental images in the population. This is usually called his ‘breakfast-table questionnaire’, because that was what Galton suggested that his subjects try to imagine.

What Galton found, to his astonishment, was that the range in imaging ability was enormous and also that there were more forms of imagery than were generally known to exist.

So whereas it was formerly thought that human imagery was all of a piece, Galton found that a small percentage of people also had 0 or no images, another small percentage had 10 or eidetic/ photographic images, and many of those who had these also had other forms of imagery for which there were then no names. One of these Galton called number forms. Another only took its present name in the 20th century, namely synthesthetic images- for example colors that are heard- as did eidetic images.

...To appreciate the difference between the two extremes of 0 and 10 images, here are some responses which Galton received from those in the lowest and highest groups. I begin with lowest:

“Extremely dim. The impressions are in all respects so dim, vague, and transient, that I doubt whether they can reasonably be called images. They are incomparably less than those of dreams.”

“My powers are zero. To my consciousness there is almost no association of memory with objective visual impressions. I recollect the breakfast-table, but do not see it.”

As for the highest, we have the following:

“The image once seen is perfectly clear and bright.”

“ . . . I can see my breakfast-table or any equally familiar thing with my mind’s eye quite as well in all particulars as I can do if the reality is before me.”

...According to present-day psychologists, roughly about 2% or 3% of the population are non-imagers. In my own study of imagery, which stretches over the past ten years, I have come across only one person who had no images whatever in his waking life—either voluntary or involuntary—although even he has imagery in dreams. Perhaps the most interesting and influential case of someone without even dream imagery is to be found in Charcot’s Clinical Lectures. This was a merchant who originally had extremely strong imagery, but who lost all of it following a crisis in his life. In a letter to Charcot he described his condition in the following way:

I possessed at one time a grand faculty of picturing to myself persons who interested me, color and objects of every kind . . . I made use of this faculty extensively in my studies. I read anything I wanted to learn, and then shutting my eyes I saw again quite clearly the letters with their every detail . . . All of a sudden this internal vision absolutely disappeared. Now . . . I cannot picture to myself the features of my children or my wife, or any other object of my daily surroundings . . . I dream simply of speech . . . I am obliged to say things which I wish to retain in my memory, whereas formerly it was sufficient for me to photograph them in my eye.

Because I have little or no imaging I find certain kinds of problems, such as those which engineers or architects or interior decorators typically tackle, extremely difficult if not impossible, since I can’t form the requisite images, let alone manipulate them... As a schoolboy I found it very difficult to read the novels assigned in English class. At the time, I believed this was because I was either not very bright or didn’t like novels... In fact, I might as well come fully out of the closet and confess that I rarely read anything apart from comic books before I was seventeen, and that I had a vast collection of comic books. But it was only when I came to work on images that I found a way of understanding this. Most novels begin (and often continue) with long descriptive passages. But such passages are very heavy going for low or non-imagers and through no fault of their own. For a poor imager, these descriptive passages are essentially just words or dense forests of words, since the weak imager cannot see anything of the scene or people painted by the words. The comic book does that work for him by means of pictures.

So how did my imagery impairment affect my philosophical work? One clear way, I believe, is that it prevented me from properly understanding the philosophers that have been the focus of much of my work- the classic empiricists, Locke, Berkeley and Hume. Of course, I knew that they put ideas at center-stage in their philosophies, and that ideas for them are very close if not identical to images. Thus Hume is clear that ideas are copies of impressions and impressions are what we experience. And since for Locke and (with qualification) Berkeley, but especially for Hume, a word is only meaningful if it stands for an idea, it follows that for them meaningful thinking consists in having an appropriate train of images. Of course, I knew all of this, but really I could never take it seriously. For how could anyone seriously believe that all their thinking was carried out in images? On the other hand, I was aware that they did seem to take that theory of meaning very seriously, particularly, in Berkeley’s case, to reject abstract general ideas and matter and, in Hume’s, to reject necessary causal connection, substantial minds and even personal identity. In short, I had a difficulty that I couldn’t really resolve. The best I could do was to suppose that either these philosophers were confused or when they spoke about images or ideas as copies of impressions or sensations, they really, in their hearts, meant meanings or concepts. Yet I now wonder how I could have been working on these philosophers for so many years, without seeing that my incredulity and difficulty stemmed from my imagery deficiency and from not being able to accept that their minds worked differently from my own. Curiously, however, I remember that when I first encountered Berkeley as an undergraduate, I became aware of the difficulty, but, after some initial perplexity, assumed that I must be wrong to be puzzled by it, since no one else seemed to be. Now, however, I am prepared to accept that the empiricists meant what they said, because they had unusually strong imagery.

If my experience is anything to go on, what mainly stands in the way of this acceptance is not only the rarity of such strong imagery, but also the natural tendency that each of us has to suppose that what is mentally normal for us is just normal or normal for all human beings. This tendency does not operate in the bodily realm because there the differences are apparent. Hence it comes with a sense of shock for either a very weak and very strong imager to realize that he is unusual. But this only emerges when a discussion can be moved from the general to particulars and concrete details.

Both Galton and James were aware that there were philosophical implications flowing from the findings of the breakfast table questionnaire—the most central of which was that philosophers had hitherto wrongly assumed that there was only one form of imaging... Very briefly, T.H. Huxley was the first to comment on the dispute, although his focus was on Berkeley’s position and Hume’s support for it, which Huxley disputed, at least for sensible or natural objects, on which he was clearly drawing on Galton’s findings both on imaging and photographic work on generic images. James, after quoting Huxley at length, then suggested that the dispute between Locke and Berkeley might be resolved if we looked at the differences in their imaging powers. A.C. Fraser took this one step further in 1891, suggesting it was Berkeley’s and Hume’s exceptionally strong imaging powers, connected with their relative youth at the time, that encouraged them to believe, as against Locke, that there were no general ideas. Put in another way, their imaging power were so strong and detailed that it psychologically prevented them from forming vaguer general images; whereas Locke, being older and hence a somewhat weaker imager, could form such images.

This brings me to the wider implications of Galton’s discoveries adverted to by James: that the variations in imagery powers show that there are basic differences in the way that human beings think, and hence that the idea of typical human thinking or typical human mind is a fiction. Following James, I have elsewhere described this as the Typical Mind Fallacy—or TMF for short. Put in another way there is no uniformity in our thinking and hence no uniform or typical human mind. As Galton put it in his Inquiries:

It will be seen how greatly metaphysicians and psychologists err, who assume their own mental operations, instincts, and axioms to be identical with those of the rest of mankind … The differences between men are profound, and we can only be saved from living in blind unconsciousness of our own mental peculiarities by the habit of informing ourselves as well as we can of those of others.

...Thus when a philosophical debate has reached a deadlock situation, or when philosophers find themselves repeating the same assertions and mounting the same arguments again and again, and to no apparent purpose, then something else should be tried. I would say it is time to bring in the Philosophical Counselors for these philosophers... In short, philosophers should go for Philosophical Counseling in the way that some husbands and wives eventually and reluctantly decide to go for marriage counseling—when they no longer seem to have any common ground, when using words and arguments no longer seems useful... Locke should not say to Berkeley: You are wrong about abstract general ideas. The reason why you disbelieve in general ideas is because your strong photographic imaging incapacites you from having general images. The idea is that going to a marriage counsellor or Philosophical Counselor is not for the purpose of deciding who is right or wrong, but easing the conflict and bringing out the hidden sources of conflict, which (we suspect) lie in the minds rather than the theories of the disputants. So here the Philosophical Counselor is not looking at the truth of theories. That is not his province. In the matter of rightness or truth, the philosopher per se should decide.

...This is the way I read Gilbert Ryle’s famous or infamous rejection of mental images in chap. VIII of The Concept of Mind. We can admire Ryle’s defence of his position as a tour de force of philosophical ingenuity. But, as we know, by positioning himself behind the barricade of linguistic analysis and argument, he was actually preventing himself from seeing the truth that some minds can form mental images. He was also adding another example to what is sometimes called the scandal of philosophy, the intractable disputes that, unlike science, populate our discipline. Not more argument, but a change in attitude or empathetic stretching assisted by Philosophical Counseling, is required.

In this respect the work of the Philosophical Counselor, as I understand him, is like that of the depth psychologist. Both are different from most other practitioners, such as dentists, who can treat their patients without ever having made their particular minds the object of a similar study. Here the Philosophical Counselor seems close to Socrates’s concern to know himself. For the Philosophical Counselor must be aware of his own cognitive capacities, if he is going to help others to understand theirs.

trade and monetary imbalances

Monetary policy can not be determined independent of trade policy.

Here is an example where a trade imbalance between two parties leads to lower interest rates in both parties. An asymmetrical situation in one domain results in a symmetrical outcome in another.

There is a widespread misconception that the United States relies on the savings of other countries to finance its current account deficit. This is incorrect. During recent years, at least, the U.S. current account deficit has been financed primarily by money newly created by the central banks of other countries. Newly issued paper money is not the same things as a country's savings. The companies that earn money by exporting to the U.S. keep their savings. It is only that they keep them in their domestic currencies after having sold the dollars they earned from exporting to their central bank. In fact, the banking systems of the export-oriented economies all across Asia are burdened by too much savings. Deposits are accumulating in the banks more quickly than there are viable lending opportunities and, consequently, interest rates have fallen to historic lows...

Many countries around the world accumulate large stockpiles of dollars as a result of their trade surpluses with the United States. The central banks of those countries print their own currency and buy those dollars in order to prevent their currencies from appreciating when the private-sector companies that earned the dollars exchange them for the domestic currency on the foreign exchange markets. The central banks then invest the dollars they have acquired into U.S. dollar-denominated debt instruments, preferably U.S. Treasury bonds or agency debt, in order to earn a return. If the amount of dollars accumulated by foreign central banks exceeds the amount of new debt being issued by the U.S. government and the U.S. agencies during any particular period, then the central banks will buy existing government and agency debt instead of newly issued debt. By acquiring existing debt, they push up the price and push down the yield.

Duncan, The Dollar Crisis (Page 298-300)

Here are several clarifications. (Let's use China as an example.)

1) US interest rates go down basically because dollars circulating inside the US real economy gets drawn into the US financial markets.

2) Chinese goods get consumed by Americans, and Chinese exporters of these goods now hold newly minted renminbis. Once these new renminbis get deposited at local banks, Chinese interest rates experience a downward pressure.

3) Continuing trade imbalance forces the Chinese central bank to print more and more renminbis. Once the lending opportunities become present, this accumulation of printed money generates either an inflation in the Chinese real economy or an asset bubble in the Chinese financial economy.


Here are two explanations of how a country, which adopts export-oriented policies and pegs its currency to dollar, imports inflation from US when the dollar depreciates.

1) As renminbi depreciates along with dollar, Chinese exports become more competitive in the world market. More foreign currency gets deposited at the Chinese central bank as exporters sell more of their goods to foreign customers. For the acquisition of these incoming foreign currencies, the central bank prints more renminbis. Now that more local currency is circulating around the Chinese real economy, inflation picks up.

2) As renminbi depreciates along with dollar, imports become more expensive in terms of renminbi. (It is tough to tell whether this development will result in a decrease in the amount of local currency circulating. An imported good may have a close local substitute. Hence an increase in the price of the former may result in an increase in the demand for the latter. So some renminbis that was destined to be converted to dollars may now stay within the local economy and increase the amount of local currency circulating. On the other hand, some consumers may continue to buy the imported good at the increased prices. That means more renminbis may get deposited at the Chinese central bank then the usual. In other words, the amount of local currency circulating may decrease.)

categorical metaphysics

Say the universe is a category. It is made of some objects whose existence can be perceived by us only through their interactions. A morphism from an object A to an object B contains the information on how A "acts" on B. If this indeed is the universe, what could the perceived universe be like?

1) We can only perceive relations between objects. For instance, the stationary and non-interacting objects is forever undetectable by us. In other words, we only perceive the arrows of the category, not the objects.

2) Moreover, we can not perceive all the arrows. A certain collection of objects and their interactions with each other were irrelevant for our evolutionary history. Hence they are beyond our sensory realm. In other words, there exists a full subcategory that lies outside our perceived universe. Call this subcategory F. (Here "F" stands for the word "forbidden".)

3) The arrows we can perceive do not form a full subcategory. We can detect only some of relationships between A and B, for the same reason as in 2.

Category theory focuses on arrows. Objects are black boxes. This is epistemologically sound. You can not know anything inside out. You do your inquisition from outside in.

Note that the above observations have a nice corollary. We can divide the objects outside F into two classes. Those which are on the periphery P of F and those that are not. Those which are in P have at least one non-empty morphism set with an object from the forbidden subcategory F. (In other words, objects outside F and P have no relationships with F.) This implies that, in our observable world, we may witness actions that emanate from F. (e.g. Observational effects of mirror matter)

fuel optimization

1. Observation

One day I decided to drive to work in a different fashion. Instead of trying to minimize the amount of time spent on road, I made an effort to minimize the amount of fuel consumed. I did not take any short-cuts. I just followed the usual 100 km long route. I managed to bring down my fuel consumption from the usual 11 liters/km to 7.7 liters/km. This was not the surprising element of the trip though. What amazed me was how many others were also driving with the same goal in mind. It was quite wonderful to look at the rear mirror and see how synchronously all the drivers were taking their feet off the gas on hilltops. We have a tendency to believe that others are like us. We assume that they are maximizing the same parameters in order to reach the same goals. The only way to feel the presence of diversity is by stepping out of the routine and gearing into an exploratory mode.


2. Gozlem

Evlerine donerken benzin optimizasyonu yapanlara aciyorum. Futursuzca gaza basip, gereksiz frenler yapanlar icin mesai saati ofislerinden ciktiklari anda biter. Surekli optimizasyon yapan ya da yapma gerekliligini hisseden birey ise ozgur degildir. Bu tip kisiler ofislerini terketmis olmalarina ragmen islerine devam etmektedirler.

zebra shit

While I was in Namibia, I had one of those experiences which under normal circumstances would be passed as a mundane event.

It was the hottest moment of the day. We were cruising in a safari Jeep. Animals were not hard to spot because they all had taken refuge under the trees which were scattered around the landscape in small bunches. Oryxes and zebras used the shadows for their comfort and survival. In return they gave something back to the trees... Something quite vital... Their shit of course!

Trees were obviously happy with this trade. But, wait, this was not an actual trade. Zebras were giving away their shit for free! They were blissfully unaware of the nutritional benefits of their shit. Trees, on the other hand, could not choose to reserve their shadows for animals who would graciously leave their shit behind.

So how did this state of affairs emerge?

Here is one speculation: On a flat, dry and infertile land, animals that obstinately avoid shitting under the trees will not be able to survive the summertime there. Without the additional nutritions trees will not grow too much. In fact they may not grow at all. Subjected to high temperatures and deprived of cool shadows, animals will experience constant perspiration. They will dehydrate, and be forced migrate to wetter lands.

necessary illusions

It is surprising how little I am involved in the perpetuation of what I call "myself". At the moment, I am passively lying on this bed, hoping that my body will fight off the disease and that I will be able to roam free again. I witness my temperature go up and down. I keep losing and regaining my appetite. I say "my" temperature and "my" appetite but none of these things are really under my control. In fact, what I rationally desire often conflicts with what my body dictates. For instance, whenever I regain my appetite, I try not to eat anything because I feel afraid of throwing up again.

Anyways... These are necessary illusions. Once I recover, I will again forget how little I am in control of things.

linear regression

Following is taken from a conversation I had with an MBA student whose background was engineering.

MBA: People at this business school mistake linear regressions for models. When I was in college, we used to analytically solve complicated differential equations in order to model motions of various systems. That was some advanced mathematics, some real modelling... This regression stuff on the other hand is a joke.

Me: Use of advanced mathematics does not imply that there is some advanced modelling going on. Sometimes the intuition required to discover a pattern will require immense amounts of intellectual effort and creativity, but the final mathematical description of this pattern will be fairly elementary. This indeed was the case with Einstein's theory of special relativity.

Complicated mathematics often lead to decreased conceptual clarity. Hence the reason why mathematicians struggle to find simpler and more conceptual proofs for results whose validity have been demonstrated via unnatural and complicated means.

You probably derived those differential equations by applying Newton's laws to not-too-complicated systems. Strictly speaking you did some modelling while choosing which assumptions to make during the simplification of the system in question. However the real modelling was done on a more basic level by Newton himself.

Newton could not have directly intuited the dynamics underlying the motion of a complicated system. Human mind does not work that way. He started out with simpler systems and modelled them first. The principles he extracted turned out to be universal and applicable to more complicated systems in a well-defined manner.

Unlike physics, macroeconomics does not lend itself to ground-up modelling. Economists have tried to aggregate models of individual level decision making processes to an aggregate level dynamical framework. None have achieved any success.

Say you have two vectors of data and you suspect that there exists some kind of a statistical dependence between them. You create a scatter-plot and look for a functional relationship between the two vectors. The technique of linear regression amounts to drawing the best fitting straight line through this data set:

You are of course free to fit anything to the data. You could try approximating it with higher degree polynomials or trigonometric series. The simplest thing to do however is to approximate it using a linear function. Remember that you do not want to over-analyze. "Is there a linear relationship?" is the humblest question you can ask to your data. A sixth degree polynomial relationship is more likely to turn out to be spurious than a linear one.

In the future, analytical solutions to tough differential equations will not be sought after by engineers. Numerical approximations given by super computers will suffice for all practical considerations. (The algorithms at work will be quite basic in comparison to the sophisticated mathematical tools employed for extracting analytical solutions.)

In a linear regression, simplicity arises due to our lack of knowledge of the interior workings of the system and due to our goal of making robust claims about the future behaviour of the system. In Newton's case, however, the source of the resulting simplicity is a mystery. Why does nature, at certain scales, behave in such a mathematically simple fashion? We do not know.