Is Cantor space the injective continuous image of Baire space?

Is Cantor space the injective continuous image of Baire space? This is an intriguing question, since its answer depends on which axioms one adopts.

First of all, in classical mathematics (CLASS), the answer is yes. However, in intuitionistic mathematics (INT) the answer is no. Then again, in recursive mathematics (RUSS) the answer is a strong yes, since in RUSS Baire space and Cantor space are homeomorphic.

For CLASS I’ve tried to find references to the above question in publication databases and with Google, but I came up short. Many texts prove that any uncountable Polish space contains an at most countable subset such that is the continuous injective image of Baire space. It is easy to show this for Cantor space, but what if we drop altogether? Well, it is not so difficult to constructively define a continuous injective function from Baire space to Cantor space which in CLASS is surjective (whereas in INT surjectivity can be proven to fail for all such functions). I would be surprised if this has not been done before, but like I said I cannot find any references. Therefore let’s call it a theorem:

Theorem (in CLASS) Cantor space is the injective continuous image of Baire space.

Proof: We constructively define the desired injective continuous function , using induction. will send the zero-sequence to itself. The -image of other sequences starting out with will be branched off from the zero-sequence at appropriate `height’.

To this end, we inductively define on finite sequences of natural numbers. denotes the sequence of length . For finite sequences we let denote the concatenation. For any let denote the finite sequence formed by the first values of (for this is the empty sequence).

Let be the bijection from to given by . Then for we have .

For put . For put . For put .

For induction, let be a finite sequence not ending with and suppose has been defined. Then for put . For put . For put .

Finally, for let . It is easy to see that is as required. (End of proof).

Clearly, even in CLASS the inverse of is not continuous (otherwise we would also have that Baire space is homeomorphic to Cantor space!). This clarifies why the constructively defined fails to be surjective in INT and RUSS, even though in INT and RUSS we cannot indicate in such that for all .

Consider the recursive sequence given by if there is no block of 99 consecutive 9’s in the first digits of the decimal approximation of , and else. We see that is in but with current knowledge of we cannot determine any such that (go ahead and try…:-)).

In INT we can easily prove:

Theorem: (INT) There is no continuous injective surjection from Baire space to Cantor space.

Proof: By AC₁₁ such a surjection has a continuous inverse, which contradicts the Fan Theorem. (End of proof)

Now in recursive mathematics (RUSS) the Fan Theorem does not hold, and Cantor space has an infinite cover of open subsets which does not contain a finite cover of Cantor space. This enables one to define a recursive homeomorphism from Baire space to Cantor space.

Interesting symmetry, since in CLASS and INT fails to be surjective, although this time in INT we cannot even indicate in for which we cannot find such that . (in CLASS we `can’ indicate such an , but this is necessarily vague, any sharp indication is necessarily recursive!). So in CLASS and INT one relies on (the intuition behind) the axioms for the statement: not all sequences of natural numbers are given by a recursive rule.

This intuition can be questioned, see my paper `On the foundations of constructive mathematics — especially in relation to the theory of continuous functions‘ (2005), and the book Natural Topology (2012).

But this post is just for fun. I wonder what happens if under (see above) we pull back the compact topology of Cantor space to Baire space…probably not very interesting but let me ponder on it.

Addenda and errata for Natural Topology (2nd ed.)

[Never my finest moments: discovering flaws in what I tried so hard to create as a perfect piece of work…:-)]

I’m rereading the second edition of Natural Topology (available at http://www.fwaaldijk.nl/natural-topology.pdf, as well as on arXiv), and naturally I am spotting some omissions, typos and even errors.

I will make a list of these in this post, and I will update this list until I replace the second edition with a third (which then in turn I fear will still be in need of a similar list, but one hopes for improvement along the line).

The most important change to be made is related to theorem 1.2.2. The given proof in A.3.1 of this (beautiful) theorem is partly deficient, because it fails to take into account the strict requirement (i) in definition 1.1.2. of `morphism’. At the time I saw no need to relax this requirement, since everything seemed to work smoothly, and in all `regular’ situations this requirement is fulfilled. So I opted for some form of aesthetic optimality.

In hindsight, I should have noted that the requirement (i), which is phrased for dots, is too restrictive in a pointwise setting. Thankfully the remedy is easy: simply replace this with the slightly less restrictive pointwise phrasing (see below), and all is well. No need even to change any other wordings, in proofs or elsewhere.

But in almost all relevant situations, the requirement (i) is easily met. I would like to mention this in an elegant way, without making a separate distinctive definition of say `morfism’ (to be pondered on). The other addenda and errata are all very minor, so far.

Errata:

(X) Definition 1.1.2. should read:

“Let and be two natural spaces, with corresponding pre-natural spaces  and . Let be a function from to Then is called a refinement morphism (notation: -morphism) from  to  iff for all and all :

(i)  is in  (`points go to points’)

(ii)  implies .

(iii) implies (this is an immediate consequence of (i))

As indicated in (i) above we will write also for the induced function from  to . The reader may check that (iii) follows from (i). By (i), a -morphism  from  to  respects the apartness/equivalence relations on points, but not necessarily on dots since does not necessarily imply  for . This stronger condition however in practice obtains very frequently.”

.

Addenda:

(O)(in the terminology of erratum (X) above):

If , then by (ii) we know that for all in . Therefore, if necessary we could `update’ the apartness to ensure …but we cannot guarantee that this is simultaneously possible for all similar pairs of dots in .

However, most spaces naturally carry an apartness on dots such that if for all in , then . In this situation, (ii) of the definition becomes equivalent with (ii’): implies for all . (This (ii’) is part of the original definition 1.2.2., which should be replaced by the above definition.)

(OO) It should be noted that is -isomorphic to . This means that it always suffices to look at .

(OOO) It should be noted that all natural spaces ‘are’ spreads already, when looking at their set of points. This is another (perhaps easier) way of seeing that any natural space is spreadlike. Let be a natural space with corresponding pre-natural spaces . Assume is an enumeration of . To create a point in , one can start with any basic dot as . Then, one chooses , and for the next values of one is free to choose basic dots , but at stage we must choose for a basic dot such that is apart from at least one of the constituents of . Then one chooses , etc. Therefore we see that, if one disregards the partial order, we did not create any new structure outside of Baire space. And there is no problem whether our point are sets (contrasting to formal topology, where there seems to be a problem in general whether the points of a formal space form a set).

We live through maps (1b): Wir machen uns Bilder der Welt (end)

[continued from the previous post, this series of posts is a complete translation of `Philosophy Paper, written by F.A. Waaldijk, student of mathematics, student number 8327661, in the year 1991′]

[—–sixth and final post in the translation—–]

Again, all this is easy to come up with. So much so even, that for what I’ve said so far, I needn’t have written down the main thesis. Maybe it gets more interesting when we look at the logos. As part of the mythos, this also consists of maps, which are also inaccurate, different in character and often incompatible with each other, in contradiction with each other. Such contradiction is usually taboo in the logos, the reason for which is not completely clear to me (which may sound strange for a student of mathematics).

This distaste of contradictions leads to the following: within the logos only those routes are acceptable, which arise in the following way:

From all maps of a certain initial subject which are at your disposal, choose a number which do not contradict each other (limit yourself to one map if necessary). These are the initial maps (premises). Reject all maps which are contradictory to these initial maps. You are now allowed to glue many other maps to the initial maps, with the condition that the maps you glue do not differ (too) much in character, and that on the way you do not obtain contradictory maps on any given subject.

Dear reader, I’m hard put to continue my narrative. Before I know it I’m busy doing myself what I detest so much, pouring ideas into words and then becoming so chained to those words that these same ideas end up pressured and stuck. May I ask of you to first read a dialogue by Plato, a dialogue between Socrates and Euthyphro, titled `Euthyphro’ ({4}, pp. 17-33)? This, it happens, is a nice story and will lighten and distract us from all that serious heavy talk, all that logos… Once more, let me write a story, a song, a poem, an essay on math education, a letter to queen Isabella, a Master’s thesis in mathematics even, but not a paper on philosophy, that exceeds my powers. Socrates asks Euthyphro[1]:

S: … And therefore, I adjure you to tell me the nature of piety and impiety, which you said that you knew so well, and of murder, and of other offences against the gods. What are they? Is not piety in every action always the same? and impiety, again- is it not always the opposite of piety, and also the same with itself, having, as impiety, one notion which includes whatever is impious?

E To be sure, Socrates.

S And what is piety, and what is impiety?

Still in that serious language for a short while: Socrates asks Euthyphro for his maps of `piety ‘ and `impiety’, but it is not an innocent question, since he gets Euthyphro to concur in advance that these maps must be consistent, compatible, not contradictory and not too different in character. Therefore Euthyphro not only has to have a small map enabling him to decide that it is pious to indict his father for murder, but also an enormous collection of maps which are suitable for every route/situation, and do not yield contradictions when glued together.

Subsequently Socrates takes Euthyphro on a long walk, too long and too difficult for Euthyphro’s small maps. Every time he asks Euthyphro: do you agree with how I glue these maps together? And: do you agree with this map? And: then you should also have a map of this now, shouldn’t you? Without fail Euthyphro ends up in trouble:

E: I really do not know, Socrates, how to express what I mean. For somehow or other our arguments, on whatever ground we rest them, seem to turn round and walk away from us.

Yet Socrates carries on, and he glues the maps together so beautifully that Euthyphro agrees time after time. In the end Socrates returns to the starting point of the walk, and to Euthyphro’s despair the first and the last map (both of the starting point) contradict each other. Euthyphro walks away in desperate confusion. He could have defended himself by remarking that all maps are inaccurate, that a competent gluer using beautiful gluework can lead any initial map to a contradiction, without reflecting on the quality of the maps!

What Socrates does in this dialogue is considered by Wittgenstein, in his Tractatus Logico-Philosophicus ({3}, 6.522 – 7), to be the only correct method of philosophy:
.

6.522	Es gibt allerdings Unaussprechliches. Dies zeigt sich, es ist das Mystische.
6.53	Die richtige Methode der Philosophie wäre eigentlich die: Nichts zu sagen, als was sich sagen läßt, also Sätze der Naturwissenschaft— also etwas, was mit Philosophie nichts zu tun hat—, und dann immer, wenn ein anderer etwas Metaphysisches sagen wollte, ihm nachzuweisen, daß er gewissen Zeichen in seinen Sätzen keine Bedeutung gegeben hat. Diese Methode wäre für den anderen unbefriedigend—er hätte nicht das Gefühl, daß wir ihn Philosophie lehrten—aber sie wäre die einzig streng richtige.
6.54	Meine Sätze erläutern dadurch, daß sie der, welcher mich versteht, am Ende als unsinnig erkennt, wenn er durch sie—auf ihnen—über sie hinausgestiegen ist. (Er muß sozusagen die Leiter wegwerfen, nachdem er auf ihr hinaufgestiegen ist.) Er muß diese Sätze überwinden, dann sieht er die Welt richtig.
7	Wovon man nicht sprechen kann, darüber muß man schweigen.

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But Wittgenstein in my opinion confuses the branch with the tree:
.

2.171	Das Bild kann jede Wirklichkeit abbilden, deren Form es hat. Das räumliche Bild alles Räumliche, das farbige alles Farbige, etc.
…
2.182	Jedes Bild ist auch ein logisches. (Dagegen ist z. B. nicht jedes Bild ein räumliches.)
2.19	Das logische Bild kann die Welt abbilden.
…
2.21	Das Bild stimmt mit der Wirklichkeit überein oder nicht; es ist richtig oder unrichtig, wahr oder falsch.

.
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Why would we accept only the logical maps, why should we be so afraid of contradictions, or maps with highly divergent characters? If we’re only allowed to walk the logically-acceptable routes, and this is a crucial point, we do not only impoverish our emotional world, but we also impoverish science, the mind, the logos itself.

The scientific method was originally a way to create new maps, which in some situations are more accurate than other, older maps; and not I think meant as a dogma that prohibits other fruitful routes. If we’re only allowed to walk the logically-acceptable routes, do we not then deprive our world of all colour, smell, etc., even our logical world? Do we not then flatten our world, where it rather is a sphere? Put differently, is that not a cramped attempt to glue the maps together in such a way that the world fits on one map, spherical or flat? And all those other maps, however beautifully drawn or how funny (humour is usually hard to find in the exact sciences), we are then supposed to reject? Or at the least distrust? No reader, I’m sorry, I prefer Márquez ({2}, pp. 1-9):

Among all those smells the smell of catholic rites dominates. There was always some saint’s day, always some procession. But there is another association which is much more complex. It is so present in my books, because it is so present in my life. For instance, I said something once which was entirely clear to me. They gave me a cup of tea, a tea that had infused for a very long time. `That tea tastes like processions’, I said. That is a very complex association. Because I don’t know why that tea for me had the same smell as the one hanging around processions. And so on…

A soup that tastes like window, a sandwich tasting like suitcase. I think everyone will understand what that means. Even if it seems confusing, and totally unrelated. A sandwich tasting like suitcase is something normal for everybody.

[—–end of Chapter one—–]

Bibliography[2]:
.

{1}	Zen and the Art of Motorcycle Maintenance. Author: Robert M. Pirsig. Publisher: Corgi (Transworld Publishers Ltd.), 1985. First edition: The Bodley Head Ltd. 1974
{2}	Nauwgezet en Wanhopig. VPRO television series in four parts [transcription]. Publisher: VPRO Publieksservice, 1989 [from interviews with Gabriel Marcía Márquez]
{3}	Tractatus Logico-Philosophicus. Author: Ludwig Wittgenstein. Publisher: Suhrkamp, 1984. First edition: 1918
{4}	Philosophical texts. Philosophy reader, published by the Science Faculty of Radboud University, 1988 [Euthyphro, by Plato]

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© 1991 F.A. Waaldijk

[—–end of paper—–]

1. This translation by Benjamin Jowett.

2. The links in this bibliography of course were not part of the original paper.

We live through maps (1a): Chapter one – Wir machen uns Bilder der Welt

[continued from the previous post, this series of posts is a complete translation of `Philosophy Paper, written by F.A. Waaldijk, student of mathematics, student number 8327661, in the year 1991′]

[—–fifth post in the translation—–]

Chapter one

Wir machen uns Bilder der Welt[1]; why, how?

Why people feel the need to express themselves, to express what goes on in their world, is not entirely clear. Some say that it is because people wish to grasp[2] their experiences, to give themselves more grip on their world. And, so they say, the only way to do so is to condense those experiences, that world, for instance in a stone sculpture, or a painting, or in words. A good painting is a good painting because it is a condensation of what you experience, what you think, what you feel. A condensation which is comprehensible at least, because it leaves out everything which in daily life makes understanding impossible.

For example, who really understands well how a city like Nijmegen functions? Who knows really well what all happens in this city, what patterns, what developments, in the social field, in the economical field, in the technological field, in nature? And how do these fields interrelate? As soon as you try to think about this seriously you start to get dizzy. It is simply too complicated. In practice it boils down to that you occupy yourself with a small piece, and that there are others who occupy themselves with the glueing together of all those small pieces, etc. In this way the city is administered. But nobody understands the totality.

When I bicycle through Nijmegen, it strikes me that I know the town primarily in the following way: I know how I must cycle from one place to another, if I want to do it as fast as possible, or if I want to encounter as little car traffic as possible, or if I want to buy some bread on the way, or… It appears there are a large number of routes in my head, from which I can choose, all according to my mood and my need. It is however not the case that I hold those routes in my head in all detail, I simply know enough to be able to make a decision on each junction, from the Berg en Dalseweg left on the Corduwenerstraat, cross the Hengstdalseweg, go up, go right on the Postweg and then on the corner with the Broerdijk there is a baker whose name I do not mention because ve would not give more than five guilders for this form of advertising, which is laughable.

When I follow such a route in my head, I only ever know small pieces (crossroads, some bends, some hills, etc.) which I glue together. And those small pieces I only know from a cycling and pedestrian perspective, by car my knowledge of Nijmegen is far more limited. Sometimes it is hard to glue those small pieces together well. Suppose I ask you for the shortest way (by bike) from the Waalkade to the Radboud hospital such that you encounter three bakers and three hairdressers (in that sequence) on the way?

Put differently, my knowledge of Nijmegen consists of an enormous collection of (small) maps. When I wish to go from one place to another, I glue a number of these maps together as well as they will allow, for as long as it takes to end up with something that promises to be a good route. But these maps are not always of the same character, some are meant for the bicycle, othes for bus and foot, some indicate the elevation differences, others the probability of flat tires (bottle bank), etcetera, etcetera. And maybe the most important is that these maps are strongly imprecise, they are rough approximations, rough condensations which at least are comprehensible since they leave out almost everything which in the real situation makes comprehension impossible.

When glueing together my small maps it sometimes happens that these imprecisions add up, and that the resulting route is not the right (or the best) one. Sometimes while cycling I find that things are just a little different from what I thought, and that I should have taken the Groesbeeksedwarsweg after all. This then gives me a new small map, which in this situation is more accurate than the small maps I already had, but which in another situation might hinder me again. All in all I would get to know Nijmegen better and better, were it not for the fact that Nijmegen itself also changes. Therefore my maps age, and I have to keep on providing old maps with a stamp `still reasonably valid’ or a stamp `really no longer valid’, and sometimes a stamp `hooray! valid again’. Apart from that, I must also continually make new maps.

You may say, reader, : why don’t you buy a city map, and every five years a new one? Good question. I’m too lazy for it, I suppose. But, so I can state in my defense, for a city map the same thing holds as for my little maps. It remains a rough approximation, a rough condensation, I will always keep encountering surprises. A city map, true to its name [in Dutch: `cityflatground’] does not indicate elevation differences. And it doesn’t tell me either at which baker’s you can buy tasty bread and at which baker’s you can buy tasty almond paste pastries, where to find the prettiest catalpa and where the Japanese cherry in bloom.

After this lengthy introduction I feel strengthened to write down the main thesis of this paper:

main thesis:

The mythos, the sum total of all our knowledge, all our understanding of the world, consists of maps. These maps are small and very divergent in character. Some are rational in nature, some emotional, some differ otherwise. Some are obsolete, some are new, some are fragrant, others visual, some verbal. Some are about spring, others about Wittgenstein. Very importantly, all these maps are imprecise, rough approximations, rough condensations, rough simplifications.

Now what happens if in a certain situation I ask myself the question: how should I behave?, or equivalently: what should I do? This means I’m asking for an acceptable route, preferably a good route, which leads me from today to tomorrow, from this situation to the next. To find that route I consult my maps of the situation in which I happen to be, and I try to glue a number of these small maps together.

This is complicated by the fact that my maps are not only imprecise, and strongly divergent in character, but also often contradictory. In a certain sense it therefore is convenient to not have too many maps at your disposal. Another disadvantage of having many maps, is that at a certain point you occupy yourself exclusively with the maps, and no longer with the world itself. When was the last time, reader, that you took the time to brush your fingers over a pine cone, to look at it, smell it, throw it in the air, play football with it, put it in your mouth to see what it tastes like?

For example, someone offers a dog a lit cigarette. Most likely the dog will reject this offer, trusting on its nose (this stinks) and its fear of fire. A human being of say thirteen summers has a more difficult job of it, vir maps could be somewhat like this: it stinks, it is not allowed by my mom and dad, that’s just what makes it exciting, it’s bad for your lungs, if I refuse I won’t belong, etc. By the time ve has taken a decision, the cigarette has burnt out (well…in a manner of speaking).

[—–to be continued in the next post—–]

1. freely after Wittgenstein ({3}, 1-1.21 ; 2.063-2.12)

2. try to take `grasp’ as literally as you can [translated from the Dutch `begrijpen’ which means `grasp’ and `understand’ at the same time]

We live through maps (0c): Mythos over logos (chapter end)

[continued from the previous post, this series of posts is a complete translation of `Philosophy Paper, written by F.A. Waaldijk, student of mathematics, student number 8327661, in the year 1991′]

[—–fourth post in the translation—–]

We have now prepared the ground sufficiently to discuss mythos and logos. We start with an explanation of these concepts by Robert M. Pirsig in his book `Zen and the Art of Motorcycle Maintenance’ ({1}, chapter 28}:

The term logos, the root word of `logic’, refers to the sum total of our rational understanding of the world. Mythos is the sum total of the early historic and prehistoric myths which preceded the logos. The mythos includes not only the Greek myths, but also the Old Testament, the Vedic hymns and the early legends of all cultures which have contributed to our present world understanding. The mythos-over-logos argument states that our rationality is shaped by these legends, that our knowledge today is in relation to these legends as a tree is in relation to the little shrub it once was. One can gain great insights into the complex overall structure of the tree by studying the much simpler shape of the shrub. There’s no difference in kind or even difference in identity, only a difference in size.

Thus, in cultures whose ancestry includes ancient Greece, one invariably finds a strong subject-object differentiation because the grammar of the old Greek mythos presumed a sharp natural division of subjects and predicates. In cultures such as the Chinese, where subject-predicate relationships are not rigidly defined by grammar, one finds a corresponding absence of rigid subject-object philosophy. One finds that in the Judeo-Christian culture, in which the Old Testament `Word’ had an intrinsic sacredness of its own, men are willing to sacrifice and live by and die for words. In this culture, a court of law can ask a witness to tell `the truth, the whole truth and nothing but the truth, so help me God’, and expect the truth to be told. But one can transport this court to India, as did the British, with no real success on the matter of perjury because the Indian mythos is different and this sacredness of words is not felt in the same way. Similar problems have occurred in this country among minority groups with different cultural backgrounds. There are endless examples of how mythos differences direct behavior differences and they’re all fascinating.

The mythos-over-logos argument points to the fact that each child is born as ignorant as any caveman. What keeps the world from reverting to the Neanderthal with each generation is the continuing, ongoing mythos, transformed into logos but still mythos, the huge body of common knowledge that unites our minds as cells are united in the body of man. To feel that one is not so united, that one can accept or discard this mythos as one pleases, is not to understand what the mythos is.

There is only one kind of person, Phaedrus said, who accepts or rejects the mythos in which he lives. And the definition of that person, when he has rejected the mythos, Phaedrus said, is “insane.” To go outside the mythos is to become insane. …

A few remarks impose themselves on me. Firstly: the caveman wasn’t `ignorant’ by a long shot, I think. Ve saw the world differently than we do, ve maybe didn’t think that the earth is a sphere, ve thought perhaps that the sun is a great fire of the spirits, of the gods; in short ve saw other connections than we do, but that doesn’t mean ve was ignorant. I am convinced that the caveman knew/realized/felt/did valuable things which we have now forgotten, unlearned, which have become impossible for us to reach, things that may be more valuable than the things we know of which ve was ignorant.

Exactly the same holds for children. What makes us consider children ignorant is that they see other connections than we, and that they are unable to defend their views. Uncle Henk is a woman. By the time Nne has learned enough language to defend this viewpoint, ve will have lost this viewpoint.

Secondly: the mythos apparently is more than just `the early historic and prehistoric myths which preceded the logos’. If I understand correctly, the mythos encompasses all values and ways of thinking that you collect from your parents and the environment in which you are raised. One consequence is that the logos doesn’t so much come after the mythos, but rather develops and exists within that mythos. I would therefore like to sharpen Pirsig’s words as follows:

The term logos, root word of `logic’, refers to the sum total of our rational understanding of the world. Mythos is the sum total of all our understanding of the world, not only our rational understanding, but also our emotional understanding and whatever other understanding may be left. Our present-day mythos, our knowledge today, is in relation to the early historic and prehistoric myths of our ancestors as a tree is in relation to the little shrub it once was. One can gain great insights into the complex overall structure of the tree by studying the much simpler shape of the shrub. There’s no difference in kind or even difference in identity, only a difference in size.

Now the logos is just a branch in this tree. To reach it one must first clamber up the stem of the tree, this is what education makes you do.The same goes for the other branches in the tree, such as our emotional world. On the day you are born you are at ground level. But your parents and everybody else are in the tree and they will pull you up. There’s nothing you can do about it, even if you disagree, because they are much stronger than you are.

Different cultures have different preceding ancestor myths, and as a result have different mythos. This corresponds to the fact that there are many different kind of trees. Clearly an oak tree differs greatly from a palm tree, and the difference can already be seen by looking at the shrubs.

In our culture the logos branch has become rather overwhelming, threatening not only to starve the other branches, but to topple the entire tree. The mythos-over-logos argument, or what I would like it to be, points out that no matter how large this branch may seem, it is not the whole tree. Also, the shape of the branch is determined by the species of the tree. In more than one sense the mythos comes first. The same of course holds for the other branches, one could equally speak of a mythos-over-passion argument.

In some cultures people rarely distinguish sharply between mythos, logos, emotions, etc. I would like to compare this to a palm tree: there is but one all-encompassing branch and it bears a good fruit (which can fall on your head just as well though).

To go outside of the mythos is to fall out of the tree (or to jump). Depending on how high you climbed, and how far down you fall, this can be a nasty tumble.

This comparison is also deficient, very deficient even, but I will leave it for now anyway. The bit about the palm tree I can illustrate a little with words of Gabriel Marcía Márquez, which he said in the television program `Nauwgezet en Wanhopig’ [`Meticulous and Desperate’] ({2}, pp. 3-11):

People say Kafkaesque, without knowing what that means. As indication of something that is remarkable or surrealistic. The same goes on for magic realism. Everything that happens in the Caribbean or Latin America, or what is strange and unusual, they call magic realism. In fact there is no magic realism in literature. There is however a magic reality, which you can find in the Caribbean. For that you only have to go out in the streets. With that magical reality, we grew up. But that magical reality is also present in Europe and in Asia.

You however are hampered by your cultural development. All Europeans in the end are Cartesians. They reject everything that does not fall within that rational thinking. One goes to Europe, and sees things happen which are as exceptional as what happens in our countries. Moreover you could say that many of those magical influences have reached us from Europe, Asia or Africa.

We are ultimately a mixture of all of you. Everything therefore has a basis in reality. Only, we surrender to that reality more easily. We are part of it, we accept it. Your system of thought forces you to reject that reality. You have done well in life. But I believe you have less fun than we do.

Blistering barnacles, ve is right! I would be having a lot more fun if I could simply write a story, instead of a philosophy paper. But now that I’m on the job I will finish it. To end this chapter, at last we arrive at what this chapter was begun for, also fulfilling a `promise’ made in the preface:

Our belief in logos, the sum total of our rational understanding of the world, is not more than a belief. Our belief in logic and in the scientific method is not more than a belief [`belief’ also means `religion’ in Dutch], and a dangerous one if practised unconditionally, since it then threatens our other ways of understanding the world. Our belief in the word (logos) is not more than a belief, and a dangerous one if practised unconditionally, since it then threatens our non-verbal worlds. 

One who says that the world obeys the laws of logic or even of science, or even of the word, confuses the branch with the tree. 

[—–end of chapter, the next and final chapter starts in the next post—–]

We live through maps (0b): Mythos over logos (continued)

[continued from the previous post, this series of posts is a complete translation of `Philosophy Paper, written by F.A. Waaldijk, student of mathematics, student number 8327661, in the year 1991′]

[—–third post in the translation—–]

Would that God gave this not to be a philosophy paper. Then I would just write down the above conversation, nothing more, without commentary that only serves to wall up the issue. Compare this to Nne’s fist answer: `Daddy’. For hav this is amply sufficient. When pressed ve is willing to clarify this answer to: `Daddy is a daddy’, but that is already a mutilation, ve rather means: `Daddy is daddy’. Notice, dear reader, that against my wish I nonetheless am giving a commentary; forgive me, I must.

The proposition `Daddy is a man’ is not only uninteresting to Nne, it is even positively incorrect. To Nne, daddy is not a man, and Nne thinks uncle Henk more resemblant of aunt Josje than of daddy. What will happen to Nne? Ve will learn that in order to be rid of the harping [on correct gender], ve must split people in two categories, where someone from the one category is a she, a woman, and someone from the other category is a he, a man. Ve will learn how to make this classification using several recognizable (but not necessarily more noticeable than non-gender-based) differences such as long hair, a low-pitched voice, a skirt, a beard, etcetera.

It may be obvious that Nne, if ve doesn’t learn to make this distinction, will hardly be able to stand vir ground. It will also be clear that this learning process profoundly influences Nne’s cognition [denkwereld; `thinking’]. Ve is forced so to speak to adopt age-old thinking patterns, even before ve can reasonably agree or disagree, in this case the thinking pattern that of all the differences between human beings, the woman-man difference is the most important. A remarkable thinking pattern, because it says that the mommy and the daddy of Nne, who differ two years in age, have the same complexion, the same way of speaking, the same cultural background and whose outlook on life is largely overlapping, stíll resemble each other less than Nne’s daddy and a twenty years older pygmy man.

Reader, I ask you in all gravity whether you accept this thinking pattern. And if you accept it, do you then accept it whilst at the same time feeling free to also reject it if necessary? Or is there something in you that immediately revolts against the idea that it is often completely irrelevant whether a given person is a woman or a man?

I ask you this last question to make a comparison. Suppose someone visits you, completely enthusiastic, almost overjoyed. Of course you ask after the reason for vir excitement. Ve answers that ve has just made a phenomenal discovery, you won’t believe it, ve says, but the earth is flat after all!

Do you think you will be able to take this person seriously for even one moment? That you will reject vir discovery is something I don’t doubt an instant, I do so myself as well. But I’m not asking you whether you are willing to reconsider the idea that the earth is a sphere, but whether you are able to reconsider that idea. You will perhaps answer that it is meaningless to do so, the flatness of the earth being a terribly outdated medieval idea. Then I make another comparison:

Suppose someone visits you, completely enthusiastic, almost overjoyed. Of course you ask after the reason for vir excitement. Ve answers that ve has just made a phenomenal discovery, you won’t believe it, ve says, but the earth is a sphere!

Do you think you will be able to take this absurd idea au serieux for even one moment? That you will reject vir discovery is something I don’t doubt a second, I do so myself as well. Nevertheless someone came to visit me the other day, with the tale that ve is going to sail to the Indies, in westward direction! Ve asked me for a small contribution to this foolhardy expedition. Ve tried to convince me that, if the earth is round, ve could reach the Indies also by sailing westward. I ridiculed hav, but ve was used to that, ve said, and whether I for once would be willing to open myself to a new idea. There ve had the best of me of course, since that is precisely what I always tell others. So I saw myself necessitated to give hav a golden dubloon. Well, ve left yesterday with vir ships. Aside from the money, I hope ve will be able to turn them around when he reaches the edge, because he is rather a nice guy. I did ask hav how ve had acquired that strange idea. To you I dare to tell this, ve said (although only after having pocketed my dubloon):

The other night my wife and I were lying in bed, and you know how loquacious my wife is, and what strange ideas she can have. Anyway, I had trouble falling asleep, and to cheer me up a little and prepare me for dreamland, she started telling me all kinds of fantastic and impossible things. She told of a Moorish wizard who cursed people to the moon. And those people, she said, had the greatest fun since the moon turned out to be a very large ball on which you could walk around, and if you continued in one direction for some time then you returned at your point of origin!
 
You will understand why I love my wife so much, she is still so, well…so childlike I would almost say. But anyway, to cut things short, that same night I dreamt that the earth is also a great ball. The next morning I told my wife what I had dreamt, and she asked me why such a thing should not be possible. What reason do you have, so she said, to assume that the earth is flat, other than that she appears flat and that everybody always says she is flat? Likewise with some Balkan tribes there are girls who decide to become a man. They flatten their breasts with tight-wound rags, and henceforth dress as a man, act like a man. They move to a region where no one knows them, and then are taken for a man, some even marry a woman! Believe it or not, their deceit is often discovered only on their death, when for that reason they are disrobed. Of course, this doesn’t occur very often, but one occurrence is already enough.
 
Now with my wife I never know whether she makes something up on the spot, or whether she has heard it from one of her many trustworthy sources. That stuff about those Balkan girls I’m therefore not so sure of, but it doesn’t matter anyhow. The earth is a woman! She appears flat but with my journey I will strip away her clothes and then we shall see!

 

he said mischievously.

When ve had narrated me this story, ve had me sold of course. Against all of my cautious nature I then had three ships built for him. When I told him this, he embraced me half in tears. You have no idea, ve said, how often I have been mocked, ridiculed, even menaced. You have no idea what this means to me. I told hav that even so I had little trust in vir undertaking as well. But you at least give me a chance, he spoke, and he really seemed deeply moved. I only hope that he will return.

[—–to be continued in the next post—–]

We live through maps (0a): Chapter zero – Mythos over logos.

[continued from the previous post, this series of posts is a complete translation of `Philosophy Paper, written by F.A. Waaldijk, student of mathematics, student number 8327661, in the year 1991′]

[—–second post in the translation—–]

Chapter zero

Mythos over logos.

Before I really begin a number of new words remain to be clarified. It concerns the word `ve’ [`vij’] and the appropriate declinations `vir’ [`vier’ en `viere’] and `hav’ [`hav’].

In Dutch [and partly English], we almost always need to specify the gender of the person or the object under discussion. `He came to stand next to me’, `I saw her walking across the street’, `have you called him already?’, etcetera. Often this gender is completely inconsequential, what does it essentially matter whether the baker is a woman or a man? The most important is that ve is a baker! At least in my eyes.

In these situations I wonder if we can effectively combat gender-based discrimination, when we perceive the world so strongly in woman-man terms that we always have to describe any given person firstly by gender. I myself support the idea that women and men are firstly human, irrespective of their gender. I also believe that the continuous stressing of the gender difference promotes gender-based discrimination [1]. A few years ago I therefore came up with some new words which can replace the traditional `she’ and `he’, in situations where gender difference is irrelevant:

she   [zij]                             he    [hij]                          ve   [vij]
her   [haar, haar (mv.)]          his    [zijn, zijn (mv.)]        vir   [vier, viere (mv.) [2]]
her   [haar]                           him   [hem]                     hav  [hav [3]] (also havself)

This is also a convenient way [in Dutch] to once and for all put an end to problems like `de Waal en haar? zijn? oevers’ [in English this problem doesn’t occur: `the Waal and its banks’]. From now on this becomes `de Waal en viere oevers’. And similarly: `het parlement en viere bevoegdheden; de gemeenteraad en vier voorzitter’ [`Parliament and its competencies; the city council and its president’]

Let us utilize this subject to address a much deeper issue. I introduce you to one of my acquaintances, a child called Nne. Nne can already talk a little. Vir aunt, aunt Josje, is visiting and coffee is served. Nne also wants coffee, and makes this known by saying:

– Me too coffee, me too
– No, Nne, that won’t do. Coffee is for grown-ups. Coffee is bad for little children. Would you like some delicious lemonade?
– But hé also gets coffee! I want too, me too coffee.

The grown-ups chuckle about Nne, who doesn’t understand this and thinks it mean.

-Listen, Nne. Aunt Josje is a woman so then you don’t say: hé also gets coffee, but…? shé also gets coffee. More importantly, Aunt Josje is big, she is grown-up. When you get to be big you can also drink coffee.

Nne therefore has to content havself with lemonade, and as a reward for doing that ve may enjoy some more education.

– Say, Nne, daddy is a…?
– Daddy
– Yes it’s about daddy. Daddy is a …?
– Daddy is a daddy.

Again laughter.

– Nne, daddy is a man. A man. And mommy? Mommy is a …?
– Woman
– Very good Nne. And aunt Josje is a …?
– Woman
– Very good. And uncle Henk is a …?
– Woman

Laughter again.

– No, Nne, uncle Henk is a man. Look, he has a beard. And he can talk in a really low voice. Henk, say `hoohoohoo’ very low, won’t you?
– Hoohoohoo.
– And you, when you get to be big, you will be a …?

But now Nne really has had enough. Ve walks away with vir glass of lemonade to more interesting places.

[—–to be continued in the next post—–]

1. `To discriminate’ actually means: `to differentiate between’

2. The objection that `vier’ already means something else [in Dutch] lapses when you compare it to `haar’ en `zijn’

3. Thought up by Reinier Post as improvement of `haam’. [in Dutch a construction like `havself’ is unneccessary.]

We live through maps (-1): Preface

[see the previous post for background; this series of posts is a complete translation of `Philosophy Paper, written by F.A. Waaldijk, student of mathematics, student number 8327661, in the year 1991′]

[—–first post in the translation—–]

Preface.

I don’t know if the hereinafter can be seen as a philosophy paper; and to be honest, when I started it I didn’t much care. I was already glad enough to find myself wanting to write down something of my thoughts, to create something. If this writing fails to meet the requirements, I will try to fashion a second essay. But in this essay I will give myself free rein, I’m almost completely worn out from the obligation to cast everything you do in an easily recognizable form, preferably a standard form. I also do not feel the need to create an overwhelming exposition, a brilliant reasoning as sharp as a sword, although perhaps I would like to regain the belief that such reasonings are more than a pleasant-or-not word play, and that it is actually worthwhile to occupy yourself with such.

Are there avenues of thinking which achieve what other avenues of thinking do not: create perspicacity, clarity, a sort of inner peace? Or is that ultimately not the domain of thinking avenues, is that the domain of: good food, sleep, good friends, woman, man, hiking, gardening? When should you stop asking yourself questions?

At this moment an oblong browngray moth is walking on the whitewashed wall. In the middle of the wall it* now pauses. If you look closely, you see a landscape of irregularities in the whitewash which the moth has to surmount, like spread-out whipped cream.

The moth disappears in the pile on top of my table, the pile of small to medium-sized objects with which I don’t know what to do. My storage space has a certain structure which doesn’t tolerate these objects: they are out of place in all my cupboards and closets. Therefore this pile remains untidied, only, in order to be able to write, I have shoved it* against the wall.

A  year ago, when I had just moved in, my table was bare. But you know how it goes. Always more new stuff and not being able to throw out the old. An extra closet might provide a solution; but for how long?

Similarly, it is hard to throw out a thought that has nested itself in your head, even though thoughts often don’t tolerate each other and you feel the need for extra heads. Maybe in general I’m hard put to see the use in philosophy because I sense too many contradictory reasonings perambulating like ghosts; out of every closet comes another thought, another norm, another sense of duty, such that the question: `what should I do?’ brings forth a many-headed howling. And you start to play them off against each other, this ghost bound by that ghost, this reasoning refuted by yonder argument, etcetera. But the pile of thoughts which you don’t know where to leave is growing.

In this essay I wll try to describe something of the pile of thoughts on my table. Even I wish to describe an extra closet where these thoughts, and others, can find a place. It remains the question whether this all will succeed. In any case the reader should not expect a straightforward [rechtlijnig, `rectilinear’] exposition. I hope to even make plausible in this essay that it is not unwise to be substantially suspicious of language in general, and of straightforward [rechtlijnig, `rectilinear’] expositions specifically. This endeavour (the making plausible that, etc.) is hampered since in this essay I must avail myself of exactly that same language.

Here you see my feelings regarding this essay. Socrates had the bad luck that his words, which he did not want to trust to paper, were written down anyway by Plato. I have the bad luck that I must write this essay. And you, reader**, have the bad luck of reading it. I hope you enjoy it nonetheless!

*[translated from `vij’ and `hav’]: see chapter zero [next post]

**[translated from `lezer’ which is gender-specific in Dutch]: here and in all of the essay is meant: reader (f/m)

[—–to be continued in the next post—–]

We live through maps (-2): perception and information

In 1991 I had to write a paper on philosophy, as part of my Master’s degree in mathematics. I wonder if philosophy is still such a prominent part of the mathematics curriculum in Nijmegen (or elsewhere) nowadays?

At the time I thought the assignment was too hard on a student of mathematics, or at least on me since I tend to take philosophy very seriously. If the great philosophers were hard put to express their thoughts in words, how could such a thing be expected from a student of mathematics? It seems comparable to asking students of philosophy to write a paper on algebraic number theory.

Yet, even though the assignment cost me a very big amount of time and energy, far disproportionate to what the teacher assumed would suffice, in the end I was satisfied with the result. And the strange thing is, after more than 20 years I am still satisfied with the result.

The paper is not too long, and written in Dutch, in a literary fashion, on a typewriter dating from the ’60s. I have decided to translate it into English for this blog, so that its contents will be findable on the web.

Before I begin, let me summarize (in hindsight) the key element of the paper:

We live through maps

In explanation of this sentence, a too short summary of the paper:

Our world is what we perceive it to be. What we perceive is determined by the information we accord to aspects of our world. This information is stored in maps, which cannot be consistent across our world. 

This inherent inconsistency, combined with the personalized character of the maps involved, go a long way in explaining why philosophy in science to me appears such a tower of Babel (see previous post).

In the next posts I will `simply’ translate the original typewriter manuscript into English chapter by chapter. Comments will be very much appreciated (see the `about’ section of this blog).

Philosophy in science: the tower of Babel

The past weeks I have had some opportunity to think about philosophy in general, and also about its relation to science (specifically to physics and mathematics which tend to preoccupy me more than other disciplines).

I frequently come across (foundational) debates where to me the key issue seems to be that we have trouble understanding what we are trying to talk about. The image that comes to mind:

The tower of Babel, Pieter Brueghel the Elder

Therefore I’m baffled that Stephen Hawking declared that `philosophy is dead’. In fact, such a viewpoint to me illustrates the need for more philosophy in (foundations of) physics and mathematics.

On what do we found our conceptions? What underpins our underpinnings? Many scientists, even in foundations, seem reluctant to really address these questions. And yes, I believe that the publish-or-perish culture is a large contributor to this reluctance.

Another contributor is the inherent hardness of addressing these questions. And then, when talking about our axioms, our basic assumptions, there is the added complication that we do not really speak each other’s (scientific) language. Even in mathematics alone, this occurs more often than not.

So would it not be an idea to give both philosophy and communication a more prominent role in (foundations of) science?