finitary axiom of choice

In mathematics, being able to select an element from a single non-empty set S is not judged to be problematic, because the condition that S is non-empty is supposed to supply automatically an element x∈S.

Is there something fishy going on here?

What is the negation of the statement "S is empty"? Is it "S contains at least one element" or "There exists an x∈S"? The latter seems to be providing you information beyond the fact that S is non-empty, but this impression is wrong. The second sentence does not really present an element from S, it presents a label for an arbitrary element of A. In formal logic, one can not construct sentences such as "S contains at least one element". An assertion of existence needs to contain a label for referring to the existent:

You manipulate the label as if you are manipulating an arbitrary element of S. The word "arbitrary" here is very important. Since you do not know which element you are manipulating, you can not use the specific properties of individual elements. But you can still utilize their shared properties! And this is what you often desire anyway. You want to operate at a symbolic level. The ambiguity works in your favour because you are interested only in the shared properties.

Note that x itself is not literally in S. It is a new symbol satisfying all the properties associated with belonging to S. Hence, in some sense, we are voluntarily setting up a curtain of epistemological ignorance here. This is the essence of any abstraction process. Many stumbling blocks in the history of mathematics was of this nature. People could not get their heads around a certain new construction because they insisted on knowing "what" they were manipulating. For instance, zero seemed to be some sort of a mystical embodiment of nothingness, while in reality it was just a symbol satisfying certain algebraic properties.

Anyway let us go back to our discussion. So, in the finite case, Axiom of Choice is indistinguishable from the statement that there exists an x∈S. In order to see this in detail, let us recall the exact statement of the axiom:

A choice function is a function f, defined on a collection X of nonempty sets, such that for every set S in X, f(S) is an element of S. With this concept, the axiom can be stated: For any set X of nonempty sets, there exists a choice function f defined on X. (Source)

Say X consists of a single non-empty set S. Then Axiom of Choice guarantees that the set of choice functions defined on X is non-empty. This allows you to pick an arbitrary choice function f. You know that f(S) is an element of S, but you can not tell which element because you can not tell which choice function f is. In order words, you are provided with an arbitrary element of S.

Note that the distinction between an actual element and an arbitrary one becomes more blurred as the set itself becomes more arbitrary. If S is equal to {1,2,3}, then the distinction is clear. What if S is an arbitrary set with three elements? Then there is almost no difference between an actual element and an arbitrary one. Hence, in that case, the label presented by the formal statement of the fact that S is non-empty can be literally treated as an element of S.

Hence, in the finite case, axiom of choice is a by-product of the notion of a "finite" arbitrary set and "finitistic" logic. Therefore we do not need it as an additional axiom in finitistic set theory. In the infinite case, it needs to be taken as an additional axiom since we continue to work in finitistic logic. If we could form infinitely long logical statements, then again axiom of choice would be a mere by-product.

This observation really clarifies the matter. Axiom of Choice comes implicitly embedded inside the notion of an arbitrary set. The basic, fundamental problem is arbitrariness... In the case of finitary Axiom of Choice, we do not recognize its presence because the logic itself takes care of it. Of course, all that the logic does is to sweep the problem under the rug.

Abel and Galois unraveled the ambiguity inherent into every algebraic equation of degree greater than one - the ambiguity in choosing one out of several solutions (roots) of such equation. Thus, they founded a mathematical framework for the study of symmetry, arbitrariness and ambiguity which has eventually grown into the modern group theory. It may seem to a non-mathematician that only Buridan's ass would have any difficulty in choosing one of the two * from **. But try to program a robot doing this, where the two * are not conveniently positioned on a line, you can not just command: "take the left one".

Gromov - Structures, Learning and Ergo Systems (Page 56)

Choosing an actual element from a non-empty structureless set is not as simple as it sounds.You need a human being to ignite the mathematical engine! In fact, constructivists claim that even a human being is incapable of this feat unless he is provided with an actual instruction for locating an element.

Don't let the simplicity of the content of this discussion fool you. These are issues of deep philosophical importance. The difficulty of making a choice in a structureless set is precisely the first problem that a believer in an observer-independent theory of reality needs to overcome.

myth of equality

Talent is overrated in business precisely because business success requires a very well-rounded skill set. On the other hand, there are many areas where talent still rules. (e.g. dance, music, mathematics, chess, athletics) Nevertheless people like Gladwell continue to believe that one can master any subject by devoting 10,000 hours to it. This is complete bullshit. He obviously has had no teaching experience. Talent distinguishes itself really early on. It is impossible to create a master painter out of a random selected person by brute practice alone. Same holds for mathematics.

For some reason, our generation is obsessed with democratization. We are constantly bombarded with the message that we can achieve anything if we really wanted to. We are all supposedly born equal. Genes and gender are claimed to make no difference, despite the mountains of evidence that they do.

The myth of equality is more damaging to the society than the belief that only talented people can succeed. It creates unrealistic expectations and dismisses the important role played by random circumstantial factors. People everywhere are blaming themselves for being a failure. It is a widespread, sick psychological situation, partly fueled by influential populists like Gladwell.

interesting factoid

Here is a factoid that I ran into while interning at a shipping company a few years ago. To my dismay, none of the colleagues found my little discovery interesting. (The level of curiosity and intellectual drive is quite low in these sectors.)

Fact

A chemical tanker burns the same amount of fuel no matter whether it is loaded or not. (Note that this statement is completely false for trucks, planes etc.)

Some Speculative Explanations

- As a ship is loaded with more weight, its depth below the water line increases. This allows the vessel to go straight through the waves rather than riding over them, and thereby decreases the actual distance travelled to reach the destination. Despite the heavier weight being pulled, the fuel consumption does not change due to this added efficiency.

- A propeller works by pushing out water particles and thereby creating a conic spiral wave behind the ship. The top of this cone is always horizontally cut out since the water particles have nowhere to go when they reach the surface. When a ship is loaded, it propeller gets further away from the surface. In other words, the mentioned cone becomes larger, and the push generated becomes greater. This results in higher fuel efficiency.

tragic discrepancies

While financial markets are hallmarks of transparency, liquidity and efficiency, commodity markets are the exact opposite. They are the most opaque, illiquid and inefficient markets of all. Despite this fact, people running commodity companies are on average much less sophisticated and educated than their counterparts in the finance world. This discrepancy is twisted and tragic.

Dealing in commodities requires much more than a smart mind:

You need to be very proactive while gathering information. (Even something as basic as who has how much stock is usually treated as secretive information.) This includes attending long, tiresome conferences, arranging prostitutes for certain people and bribing certain others. Since governments are frequently involved on the supply side and since supply is often constrained, you also need to maintain close relationships with the relevant officials.

You have to know the players in the your sector really well. Moves and trades in financial markets are anonymous, but those in commodities can be identified and traced. (The number of important players are a lot fewer.) Hence knowing the personalities and quirks of traders will give you hints about how they may behave under future circumstances and how they may respond to your future actions. You also need to keep track of who are friends with whom. Potential off-the-record collaborations happen all the time and change the market dynamics.

• You need to watch closely certain macro economic indicators such as interest, inflation and growth rates. They affect each player's financing costs and investment projections. They also influence the demand for your products via a variety of complex channels.
• Although most markets are denominated in dollars, your costs may not be. This means that you need to hedge whatever exchange-rate risks you are exposed to.
• Finalization of each transaction usually takes some time and during this period your initial assumptions may go haywire. This requires you to manage counter-party risks via the available commodity finance and insurance mechanisms.
• You need to keep track of developments in the futures markets which may be dominated by finance companies completely unknown in the spot markets. Future and spot prices can interact in very complicated ways. Causal arrows go both ways.
• There is no single price in any commodity market. Each quotation is a function of delivery date, place and form of financing. In particular, this implies that you need to be on top of things in the shipping markets. Lots of things can go wrong.
• As I had remarked elsewhere, a typical commodity supply agreement contains many hidden options. Pricing of these options requires as much financial sophistication as the most complicated structured finance products demand.
• As if all these intricacies were not enough, in any given market, there is usually no single unique commodity that is being traded. In coal markets, for instance, traders negotiate prices with respect to parameters such as moisture, ash, calorific and sulfuric content. In chrome ore markets, they pay attention to iron-chrome ratio, and iron and silicon contents. The number of parameters often overwhelm those witnessed in the financial markets. Even if you can gather some timely information from publications and private channels, the data will often be so crude that no sort of future looking analysis can be conducted.

The most protracted recessions are caused by over investment in commodities markets. We need more sophisticated people in these sectors to avoid deep economic downturns. Colossal mistakes are made on a daily basis. The only reason why they are not noticed and corrected is because there is nobody knowledgeable enough to take advantage of them.

Unfortunately, a favourable demographic evolution is very unlikely to take place. The incumbents are rough government agencies and non-meritocratic family companies which have way too much power in their hands.

information and structure

In order to unearth an underlying structure, you need to selectively forget some of the available information. This meta-principle is manifested in so many places that it would be pointless to make a list. (In some sense the notion of cognition is based on it. No wonder why it is so ubiquitous.)

Here is a quick mathematical example. In order to assign a singular chain complex to a topological space, we collate together face maps to create boundary operators. While doing so, we lose a lot of information. (The images of these operators are quite small.) Yet enough information is encoded to recover topological invariants such as the Euler characteristic.

absurdity of self-defeat

Do you want to discredit someone or revoke a thesis? Be succinct. Use sarcasm. Aim at the underlying absurdities.

Here are three great examples where a doctrine is pointed out to be a self-defeating one:

• Economist Frank Knight described the Chicago School's positivism as "the emotional pronouncement of value judgements condemning emotion and value judgements which seems to [me] a symptom of a defective sense of humor."

• Philosopher Sidney Morgenbesser's response to P.F. Skinner (the founder of radical behaviorism): "Let me see if I understand your thesis. You think we shouldn’t anthropomorphize people?"

• Philosopher Donald Davidson's take on relativism: "The dominant metaphor of conceptual relativism, that of different points of view, seems to betray an underlying paradox. Different points of view make sense, but only if there is a common coordinate system on which to plot them; yet the existence of a common system belies the claim of dramatic incomparability." (Similarly, Auguste Comte said “Everything is relative; and only that is absolute.”)

• Philosopher Ken Wilber’s take on antihierarchy theorists:

Q: But many feminists and many ecophilosophers claim that any sort of hierarchy or "ranking" is oppressive, even fascist. They say that all such value ranking is "old paradigm" or "patriarchal" or oppressive, and it ought to be replaced with a linking, not a ranking, worldview. They're very aggressive with this point; they hurl rather harsh accusations.

KW: This is a bit disingenuous, because you can't avoid hierarchy. Even the antihierarchy theorists that you mention have their own hierarchy, their own ranking. Namely, they think linking is better than ranking. Well, that's a hierarchy, a ranking of values. But because they don't own up to this, then their hierarchy becomes unconscious, hidden, denied. Their hierarchy denies hierarchy. They have a ranking system that says ranking is bad.

Ken Wilber - A Brief History of Everything (Page 25)

A self-defeating argument arises strange feelings.

• It is like a suicidal person who feels like he should have never been born in the first place. The fact that he kills himself at the end does not mean that his life was devoid of enlightening stories. Similarly, Søren Kierkegaard said the following in Concluding Unscientific Postscript to Philosophical Fragments: "[The reader] can understand that to write a book and to revoke it is not the same as refraining from writing it, that to write a book that does not demand to be important for anyone is still not the same as letting it be unwritten."

• It is false, but nevertheless the very act of self-defeat can be illuminating. The proof of "P implies ¬P" may contain derivations of some non-trivially true statements.

• It is valuable for the same reason why every false argument is valuable: It teaches a lesson.

• It is poetic. It provides a glimpse of the fiery walls separating truth from falsehood.

• It lies on the border between existence and non-existence. It is like a particle that spontaneously produces its own anti-particle and thereby annihilates itself out of existence.

• It is like the Ouroboros, both scary and mystifying.

possible worlds

Epistemic logic is a delusional subject. The notion of ignorance can not be formalized as a uniform probability distribution over the future possible states of the world, because the notion of future possible states does not make any sense. For instance, how do these states differ from each other? You need to have a theory of everything (TOE) at your disposal to answer that question. (In other words, an ignorant person is supposed to have perfect knowledge of what is possible but no knowledge of what is coming next. What an insane proposition!) Unfortunately we do not have such a theory at the moment. In fact, the reality may be inarticulable in a formal language. So there is no reason to suppose that a TOE exists. Moreover, if it exists, it may be extremely complex and inaccessible. Even if it exists and is accessible, the totality of all its future possible states may be humanly incomprehensible.

The notion of future possible states serves an instrumental purpose in finance and cosmology, and should not be taken really seriously. Even insurance companies are aware of the entirely groundless human tendency to think of the future as a model-dependent variation of the past. That is why they always use a God clause to contractually protect themselves from the totally unforeseeable surprises which the reality is always capable of producing.

While there is no model-independent way of legitimately speaking about the future possible worlds, there is absolutely no legitimate way of speaking about all possible worlds!

Even if we had a TOE, we would never know which aspects of it are arbitrary and therefore could have been different. In cosmology, one is often encouraged to engage in thought experiments where certain fundamental constants are imagined to be different than their current values. This is not a caricature example of what we will be able to do with a TOE. This sort of tweaking is exactly all we will be able to do! Of course, a TOE will be more than just a bunch of constants. But what is arbitrary about the rest of the structure will be a complete mystery. We will just have to sit down and wait for an anomaly to take place.

- Wait. What sort of an anomaly are you talking about? The TOE is supposed to explain everything!

That is my point. There are so many things wrong with the idea of a TOE! We can not rule out the possibility of a genuinely surprising future development. (Yes, I can legitimately use the word "possible" without invoking the language of possible worlds. It is an expression of my epistemological limitations.) Similarly, what is today taken as a constant may later be recognized to be a time-dependent variable. (No, a TOE will not be a timeless theory. We can explain only what we experience.) In fact, it may even be recognized to be space-dependent, as the horizon of our observable universe expands with the passage of time.

For instance, physicists have started to take seriously the idea that the fine structure constant may not be a constant after all:

Until quite recently, all attempts to evaluate what happens to the universe if the fine-structure constant changes were unsatisfactory. They amounted to nothing more than assuming that alpha became a variable in the same formulas that had been derived assuming it is a constant. This is a dubious practice. If alpha varies, then its effects must conserve energy and momentum, and they must influence the gravitational field in the universe. In 1982 Jacob D. Bekenstein of the Hebrew University of Jerusalem was the first to generalize the laws of electromagnetism to handle inconstant constants rigorously. The theory elevates alpha from a mere number to a so-called scalar field, a dynamic ingredient of nature. His theory did not include gravity, however. Four years ago one of us (Barrow), with Håvard Sandvik and João Magueijo of Imperial College London, extended it to do so.

This theory makes appealingly simple predictions. Variations in of a few parts per million should have a completely negligible effect on the expansion of the universe. That is because electromagnetism is much weaker than gravity on cosmic scales. But although changes in the fine-structure constant do not affect the expansion of the universe significantly, the expansion affects alpha. Changes to alpha are driven by imbalances between the electric field energy and magnetic field energy. During the first tens of thousands of years of cosmic history, radiation dominated overcharged particles and kept the electric and magnetic fi elds in balance. As the universe expanded, radiation thinned out, and matter became the dominant constituent of the cosmos. The electric and magnetic energies became unequal, and alpha started to increase very slowly, growing as the logarithm of time. About six billion years ago dark energy took over and accelerated the expansion, making it difficult for all physical influences to propagate through space. So alpha became nearly constant again.

In 2003 Barrow and David F. Mota of Cambridge calculated that alpha may behave differently within the galaxy than inside emptier regions of space. Once a young galaxy condenses and relaxes into gravitational equilibrium, alpha nearly stops changing inside it but keeps on changing outside. Thus, the terrestrial experiments that probe the constancy of suffer from a selection bias... No spatial variations of alpha have yet been seen... If alpha is susceptible to change, however, other constants should vary as well, making the inner workings of nature more fickle than scientists ever suspected. 

Barrow & Webb - Inconstant

Taking the set of all possible mathematical theories as the set of all possible TOEs does not resolve the issue neither. It just pushes the problem to a one step higher level. Now we will be at a loss, because it is not clear which type of logic is the right setting for these theories. (There are many different candidates out there!)

What about the set of all possible mathematical theories in all possible logics? That will not solve the problem neither. While one can legitimately consider mathematical statements within a certain logic, one can not legitimately consider the set of all logics. What will be the parameters of your variation?

P.S. I just love the insane examples that metaphysicists come up with. They literally make me burst into laughterous tears. Here is one example where a mentally challenging concept is illustrated via a physically handicapping example:

"The second type of apparent modal truth, however, is more challenging. Julius Caesar could have had a sixth right finger which was never burnt but which could have been burnt. This involves a nested possibility, which is troublesome to actualist representationism. To reveal the nesting clearly, let us articulate the possibility in question in a more pedantic and rigorous way. The following is possible: Julius Caesar had a sixth right finger such that (a) it was never burnt, and (b) the following is possible: It was burnt." (Source)

Update (November 2011):  There now seems to be some evidence for spatial variation of the fine structure constant. Here is an extract from Michael Murphy's website. (He is one of the scientists involved in the research.)

Our results are by no means conclusive. We have thoroughly searched for other possible explanations for our results. Research science is constantly plagued by the problem of "systematic errors". These errors mimic your result or somehow destroy it. They are notoriously hard to identify. And this is what we've been looking for in our results. But we still can't find anything that explains our results besides a varying alpha! The new "alpha dipole" adds a new twist as well, making it even more difficult to understand how a systematic effect (or several) might have mimicked what we found. 

The best approach in science is to always check (and re-check if necessary) your results using different equipment and analyses. We continue to analyse new, large datasets observed with the Very Large Telescope and Keck telescope. But, as mentioned above, there's a famous saying in science: "Extraordinary claims require extraordinary evidence.". Though we are claiming something quite extraordinary here, the evidence, though very strong, is not extraordinary enough. Yet. No one should really believe that constants are varying until another type of experiment confirms the results. Possibilities for other types of experiments include making very precise measurements of the fluctuations seen on the Cosmic Microwave Background sky -- the radiation left over from the big bang. Another possibility is to measure very accurately the abundances of the elements that were produced in the big bang. But these methods have their own problems and systematic errors. But we're hoping this will improve soon! 

Probably the best chance of confirming our results is to search for varying alpha in the laboratory somehow, perhaps by comparing the ticking rates of ultra-precise atomic clocks. The current best precision is not quite high enough but atomic clock technology is improving extremely rapidly, so we may know for sure sooner rather than later.  

... I mentioned that alpha is made up of three other constants: alpha = e2/hc where c is the speed of light, e is the charge of an electron and h is Planck's constant. Both laymen and scientists alike always ask whether we have any idea whether it's c, e or h that varies. This frequently asked question has a subtle and often misunderstood answer. 

In fact, one can never experimentally distinguish between a varying c or e because these quantities are always measured in some arbitrary units like meters, kilograms, seconds etc. Consider measuring the time it takes light to travel between you and me on Monday and then again on Tuesday. Imagine that the two answers were different. What does this tell you? You might conclude that the speed of light, c, has changed between Monday and Tuesday or, equally well, you could conclude that time has slowed/accelerated or that your measuring rods (i.e. meter rules) have changed length. These three conclusions are all equally valid and can not be distinguished by an experiment! But alpha is special because it is a dimensionless combination of other constants: alpha is just a number, i.e. no units! We can therefore measure changes in alpha unambiguously.  

Some confusion has arisen recently in the literature about this question. The problem is that there exist well defined theories called "Varying Speed of Light" (VSL) and "Varying Electric Charge" (VEC) theories. For example, in VSL theories, it is indeed the speed of light that is considered to vary. But this is just a mathematical convenience: one could easily convert any VSL theory into a VEC theory! The only reason one chooses to label one particular theory a VSL or VEC theory is because that theory might look simpler (mathematically and intuitively) when considering a varying c or e. Essentially, the confusion is that the (arbitrary) names given to these theories mask their inherent duality (or triality if you include h!).

fundamental ratios

1. There are no big cells.
2. There are no small animals in the polar regions.
3. Red wines are kept in small barrels for aging.
4. Grain dust is explosive but grain is not.
5. Larger ships are more efficient.

All of these seemingly unrelated phenomena stem from one single principle: As a three dimensional object gets uniformly larger, its volume grows faster than its surface area.

Going back to the listed phenomena:

1. After a certain size threshold, diffusion of molecules across the membrane can no longer handle the amount of traffic needed to sustain the cell metabolism.
2. Small animals have more trouble maintaining their body temperature.
3. Smaller barrels provide greater contact with the wood, allowing the wine to pick up the tannins faster.
4. Grain dust is smaller, and therefore has relatively more surface area that is available for chemical reactions.
5. Most of the steel used in the construction of a bulk carrier goes towards building the hull.

felsefi geyik

Tarik: Egemenlik nasil ilk batida bulunmus anlamadim.

Umut: Egemenlik kavrami, sosyal dusuncede ilk kez Jean Bodin denen bir adam tarafindan kullaniliyor. Yoksa bizim egemenlik dedigimiz sey ilk topluluklarda da var yani.

Tarik: Hmm. Entellektuel olarak ilk batida ele alindi diyorsun?

Umut: Evet.

Tarik: Illahi kaleme almak lazim zaten herseyi :) Almazsan unutulup gidiyorsun.

Umut: :) Alsan da unutuluyosun abi, cok istisnai bir sey yazmadiysan eger. O istisnai sey de daha önce kimsenin dile getirmedigi, cok bariz bir gercek oluyor. Ama ilk kavramsallastiran sen olunca "Vaaay buyuk filozof" deniliyor. Zor bir sey ama tabi.

Tarik: Allah'tan dil surekli evrim geciriyor. 10 bin yil sonra belki su anki filozoflari kimse anlayamayacak. "Acaba su kelime ile ne demek istedi?" filan diye carcur edecekler adamlarin dediklerini. Sonra ayni seyleri yeni dillerinde baskalari tekrar kesfedecek. Bu sefer herkes onlari referans vermeye baslayacak...

Umut: :) Evet abi. Felsefenin yarısı o zaten: "Acaba su kavramla ne demek istemis?" Bizim bugün bilmem ne dedigimize adam o zaman ne demis...

Tarik: Evet, icinden cikilmasi zor.

Umut: Aynen. Bundan dolayi felsefede farkli okullar, ogretiler cikiyor zaten ortaya. Sag Hegelci, sol Hegelci... :)

Tarik: Matematikte boyle bir problem yok :) Gayet guzel anliyoruz su an Euclid'in tam olarak ne demek istedigini.

Umut: :) Degil mi ?

Tarik: Isin kotusu, Hegel gibi adamlar gidip bir de en boktan sekilde yaziyorlar dusuncelerini. Yoruma acik hale geliyor hersey. Tam siciyor ortalik.

Umut: Evet :) Ama Hegel'in olayı consistency abi garip bir sekilde. Yani zaten felsefesinin dibinde bu var: Hic bir temel yok, istediğin yerden başlayabilirsin. Sirkuler.

Tarik: Dilin ozu zaten sirkuler... Sozluk mesela sirkuler.

Umut: O da dogru!

Tarik: Dusuncelerini kelimelere doktugun anda siciyor zaten olay, otomatikman. Dokmeyeceksin. Unutulacaksin.

Umut: Wittgenstein'sin :)

Tarik: En erdemli adamlar unutuldu gitti. Boyle hirs yapmis, dunyevi adamlar hatirlaniyor. Hepsi dunyaya bir sey kanitlamaya calisan, sorunlu herifler. "En akilli benim! En dogru benim!"

Umut: Abi, her seyi aciklama hirsi zaten cok manyak bir sey yani :)

Tarik: Evet. Bilimadamlarinda da var bu. Stephen Hawking tam gerizekali mesela.

Umut: O adam resmen paraplegic olmasindan dolayi bu kadar meshur. Avantajina kullaniyor.

Tarik: En son kitabinda o kadar sacmalamis ki. Aklinca metafizige karsi cikiyor. Kendi amator metafizik yapmis. Felsefi duyarliligi sifir adamin. Fizik yapsin, sussun.

Umut: Okumadim :)

Tarik: Ben de okumadim :) Elestirilerini okudum yetti.

Umut: Artık her seyi fizik kurallariyla acikliyoruz demis ama, degil mi?

Tarik: Cok aptal sorulara, aptal cevaplar vermis. Soru: "Why is there anything rather than nothing?" Cevap: "Universe just pops into existence."

Umut: Oha, bu sorunun cevabinin metafiziksel olmaması imkansız zaten.

Tarik: Fiziksel sekilde temellendiriyor cevabini. Quantum vakumlarindan filan bahsediyor herhalde...  Yanlis geyikler yani. Bosver!

Umut: Abi, "Why?" sorusunu bilim nasil cevaplasin ki?

Tarik: "Hiclik" filan sacma kavramlar abi. Evet, cevaplayamaz.

Umut: Ancak "How?" sorusuna cevap verebilir. "Why?" diye sorarsan metafizige girersin.

Tarik: "How?" sorusunu da metafizik yapmadan yoneltemiyorsun. "How" kelimesinden sonra ne diyeceksin? Muhtemelen "How something does something?" gibi bir cumle olacak.

Umut: Oradaki "something"ler de metafizik aslinda, degil mi :)

Tarik: Evet. Birincisi "existential", ikincisi "structural". "Sadece structure yeterli." diyenler cikti simdi. Gene kafa karisikligi! Neyse :)

Umut: Structuralism?

Tarik: Ontic-structuralism! Goruyorsun, nasil da yeni terminoloji uretiyorlar hemen... Kesin biri gecmiste baska bir sekilde ayni konuya deginmistir.

Umut: Tabii ki abi :) Antik Yunan'da bile bahseden olmustur, eminim.

Tarik: Muhtemelen. Sorun su abi. Metafor repertuarimiz genislemiyor hic bir zaman.

Umut: Conceptual framework'ler degisiyor.

Tarik: Hala insaniz sonucta. Sempanzelerle az cok ayni beyne sahibiz. Yazili tarih icerisinde pek bir gelisme olmadi. Explanation is deep down always metaphorical: "Something is like something."

Umut: O da sirkuler.

Tarik: Evet, mecburen. O yuzden zaten "Why?" sorusuna hic bir zaman cevap veremiyorsun. Insaniz abi. Bir parcasi oldugumuz oyunla ilgili ne kadar sey ogrenebiliriz ki? Oyun dedim bak... O da metafor!

Umut: Pek fazla degil.

Tarik: Politik felsefe filan bu yuzden guzel iste. Kendi yarattigimiz lavuk seyleri tartisiyoruz. Bir yerlere varma sansimiz daha yuksek.

Umut: Aynen :) Etik de oyle. Artefact tartisiyorsun sonucta.

Tarik: Bu arada su olaganustu bir deney: "Then there is neuroscientist Antonio Damasio work. In Descartes Error he described a study subject whose prefrontal cortex –where we do conscious thinking and deciding – could not communicate with his limbic system – the area of the brain associated with feelings and affect. This man could not make any choices or behave rationally, because the facts had no valence. Without input on how the facts felt, they literally had no meaning." Bayagi bir felsefi geyigi cope atiyor.

Umut: :)

Tarik: Bir bu sekilde rasyonalismin agzina sicmak var, bir de postmodernist (kendi kendiyle celisen) geyiklerle agzina sicmak var...

Umut: Abi duz rasyonalist pek kalmadı gibi ya.

Tarik: Kalmadi da, duygulardan bahseden de yok epistemolojide. Takmislar empiricisme. Gorsel dusunuyorlar cunku... Neyse. O study subject'in yerinde olmadigimiz icin bilemeyecegiz tam olarak nasil bir durum.

Umut: Evet. Duygulardan eskiden daha cok bahsediliyordu. Hume filan deginiyor...

anomalous spaces

Some people think that infinite dimensional spaces are weird. That is plain wrong! Such spaces emerge as a natural setting for the rings of polynomials. Who can dare to claim that polynomials, one of the first algebraic inventions of the human mind, are weird?

It is actually the lower dimensional spaces that are truly anomalous. As opposed to their higher dimensional analogues they can be visualized!

We are the source of all anomalies in mathematics. This is a profound observation and its counterpart in physics was the source of several intellectual revolutions. For instance, why are symmetry principles so prevalent and powerful in physics? The answer is very simple: They remove the effects of our own existence and help us move one step closer to the impossible dream of an observer-independent description of the reality.