Przystanek szczęścia

Dziś rano, podczas rozmowy z jedną z matek w drodze do tramwaju po odprowadzeniu Poli do żłobka, usłyszałem wzruszającą historię.
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Rozmowa zaczęła się od nawiązania do pochwały jaką jej córka, Lena otrzymała od pani ze żłobka – że jest bardzo empatycznym dzieckiem, które zawsze stara się uspokoić inne dzieci miłymi słowami lub oferowaniem zabawek gdy tamte płaczą lub nie mogą zasnąć. Rzeczywiście, pamiętam że przynosiła zabawki Poli, podczas naszych pierwszych dni adaptacyjnych.
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Tak się złożyło, że po odprowadzeniu dziewczynek, szliśmy z mamą Leny na ten sam przystanek tramwajowy, więc chcąc być miły, nawiązałem do zachowania Leny, mówiąc że skądś to ma. Od słowa do słowa, i dowiedziałem się że matka Leny od lat zajmuje się opieką osób starszych lub z niepełnosprawnościami. O, widzi pani – prawdopodobnie nie jest całkowitym przypadkiem że pani taką pracę obrała! I co może tłumaczyć naturalne skłonności Leny.
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Przyznała że rzeczywiście gdy mieszkali w Londynie, po kilku zawodowo niestabilnych latach, podczas których zmieniała pracę co najmniej pięć razy, ostatecznie najdłużej pracowała w prywatnym domu opieki, głównie dla seniorów. Po tym jak stwierdziłem że to musiała być ciężka praca, przyznała mi rację i zaczęła opowiadać o rozmaitych wyzwaniach. Wiele osób cierpiało na zaawansowaną demencję i inne niepełnosprawności, nie koniecznie związane z wiekiem.
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Powiedziała że jednym z bardziej powszechnych wyzwań były wymogi staruszków aby zabierać ich na wycieczki do ich ulubionych dzielnic i miejscowości, lub tych gdzie sie urodzili i mieszkali. Chcieli aby im zamawiać taksówkę lub zabrać na przystanek autobusowy. Jeden pan chciał aby z nim pojechać do Scarborough, ponieważ tam się wychował (to ta miejscowość z piosenki „Scarborough Fair”), a inna pani z kolei prosiła aby ją zabrać na główną ulicę Kingston, pod Londynem, tylko zapytana na którą, nie potrafiła sobie przypomnieć.
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Niestety, takie wycieczki były wykluczone, nie tylko ze względów budżetowych i bezpieczeństwa ale i smutnego faktu że większość staruszków po kilkunastu minutach zapominało swoją zachciankę. Ostatecznie ktoś wpadł na pomysł aby przed ośrodkiem postawić ławeczkę z atrapą przystanku autobusowego. Więc gdy ktoś nalegał na wyjazd, opiekuni zaprowadzali go na ten przystanek. Idea okazała się, jakby to ująć – okrutnie skuteczna, ponieważ staruszkowie z wielką przyjemnością przesiadywali na tym przystanku godzinami, nie pamiętając o tym że już tak długo czekają a żaden autobus nie przyjeżdża.

Osobliwy początek osobliwości technologicznej

Już od kilku dekad mówi się wiele o sztucznej inteligencji i o tej tajemniczej osobliwości technologicznej gdy to sztuczna inteligencja osiągnie poziom i tempo rozwoju, które zostawi ludzkość w tyle i może nawet jej zagrozić (chleb powszedni pisarzy sciencie-fiction). Ostatnio dyskusja ta znacznie się nasiliła, między innymi w związku z zaskakującymi możliwościami oraz rozpowszechnieniem czatbota ChatGPT. Słucham i doświadczam tych rewolucyjnych zmian na co dzień i przyznam że inspirują one wiele przemyśleń, których źródłem do niedawna byli głównie Asimov i Lem. Ale muszę przyznać że to wszystko nie miało na mnie takiego wpływu jak pewne osobliwe (nomen omen) zdarzenie, które miało miejsce kilka dni temu, na moim osiedlu. Jak wiemy, hulajnogi eklektyczne są wszędzie i również wiemy jak irytujący jest fakt że ludzie zostawiają je praktycznie wszędzie. Przyznam że niekiedy nawet mam ochotę taką hulajnogę kopnąć, jako zamiennik nawymyślania bezmyślnemu użytkownikowi. Przedwczoraj idę sobie do sklepu i mijam przewróconą hulajnogę. W pierwszym momencie się uśmiechnąłem, myśląc że ktoś inny dał upust frustracji, ale chęć zachowania względnego ładu estetycznego na moim osiedlu sprawiła że postanowiłem tą hulajnogę jednak postawić. I tak zrobiłem, po czym usłyszałem, metaliczny ale mimo wszystko miły głos wydobywający się z wnętrza tej podrapanej i poobijanej maszyny, mówiący -- dziękuję!

Opis wesela wiejskiego, na które miałem ostatnio zaszczyt być zaproszony.

Była to piękna i zaciszna okolica, otoczona polami i zagajnikami, położona wśród kilu uroczych stawów. Wieczór był wyjątkowo piękny – tuż po zachodzie słońca, na pobliskim polu leżał siwy szal gęstej mgły, pozłocony delikatnymi promieniami wschodzącej pełni księżyca. Niebo było nieskazitelnie czyste, więc mimo pełni widoczne były gwiazdy. W powietrzu panował magiczny bezruch, otulony tajemniczą ciszą, akcentowaną jedynie sporadycznymi plusknięciami ryb w stawie. Jedyną rzeczą mącącą tę serenadę natury był harmider urządzony przez hałaśliwe stado ludzi, który miał miejsce w samym środku tej cudownej nocy.

Jesienna melancholia

Popijając świeżą kawę w jednej z moich ulubionych warszawskich kawiarni, na skrzyżowaniu alej Solidarności i Jana Pawła II, spoglądam przez zakurzoną szklaną witrynę na złocące się we wczesno-popołudniowym jesiennym słońcu liście pobliskich, już prawie całkiem nagich klonów, oraz na złoto-czerwone sznury cicho snujących tramwajów. Przepełniona tęsknotą piosenka Zaz, 'Éblouie par la nuit', odrywa mnie od opowiadania Kena Lin z październikowego numeru Nowej Fantastyki, i nagle nie potrafiąc się oprzeć fali nadchodzącej melancholii, wpadam w zadumę nad przemijaniem i nad żyjącym cudem tego zmartwychwstałego miasta.

ALOSZA I DRZWI

Mój szczur po prostu nie radzi sobie z pojęciem drzwi. Jak są zamknięte to je obwąchuje z ewidentną ciekawością. Gdy drzwi są otwarte to jego zdziwienie wcale nie ustaje---obwąchuje próg i framugę z zaciekawieniem; rozgląda się z widocznym osłupieniem: jak to może być, że perspektywa wcześniej nie widoczna, nagle się ujawnia? Nagła zmiana w topologii podłogi musi robić wrażenie. Ale, mimo trudności z ogarnięciem swoim szczurzym umysłem pojęcia drzwi, trzeba temu ciekawskiemu gryzoniowi przyznać godną pochwały wytrwałość.

MAJOWY WIECZÓR

W lipowym zagajniku, z nad usłanej puszystyą bielą trawy unoszą swoje główki delikatne dmuchawce, jakby chciały odgadnąć które to gwiazdy wskazują skierowane ku niebu stożkowe kandelabry kwiatów samotnego kasztanowca.

Asortyment Nieskończoności

Jest to prezentacja Powerpoint, prelekcji której udzieliłem w Otwartej Kolonii (filii Wolskiego Centrum Kultury) 20 kwietnia. Ten wykład to krótka historia pojęcia niskończoności, oraz wstęp do arytmetyki liczb kardynalnych poprzez prezentację dowodów Cantora na równoliczność zbiorów nieskończonych. Poniższa prezentacja zawiera trochę definicji oraz pomocy ilustracyjnych, ale oczywiście brakuje jej niezbędnej narracji która wyjaśnia wszystkie pojęcia oraz ich relacje.

Asortyment Nieskończoności PDF by Mariusz Popieluch on Scribd

Artykuły które napisałem, i w których tworzeniu współpracowałem: Kwiecień 2017

Najnowszy Chrysler 200: Bezpieczeństwo w dobrym stylu. 

PRZYPADEK CZY SPRYT? Jak wygrana Libratusa rostrzyga spór o to, jaką grą jest Poker?

Artykuły które napisałem, i w których tworzeniu współpracowałem: Marzec 2017

Nadchodzi Etykosfera: Czyli postęp i implikacje sztucznej inteligencji. 

Uważność dynamiczna: metoda uzupełniająca Twój trening. 

Niepewna przyszłość samochodów autonomicznych. 

Jak sprawić aby algorytm YouTube pracował dla Ciebie?

Artykuły które napisałem, i te w których tworzeniu współpracowałem: Luty 2017

Nawet jeśli historie są zmyślone, to emocje i doświadczenia są prawdziwe.

Śmigaj jak mistrz Zen.
Guide to planning your Valentine’s Day in Crawley and West Sussex in 2017.

Jak najlepsi pokerzyci na świecie dbają o kondycję? 

Moda lat 20'tych. 

Nowa ustawa hazardowa wchodzi w życie na początku kwietnia.
Tutaj powinienem podkrelić że powyższy artykuł jest mało oryginalny, i w znacznym stopniu kożysta z innych artykułów oraz cytatów.

Artykuły które napisałem, i te w których tworzeniu współpracowałem: Styczeń 2017

Czy zagłada Wielkiej Rafy Koralowej jest nieunikniona?

Jaksztuczna inteligencja zdominowała przestrzeń gier.

Pozytywny wpływ gier na funkcje poznawcze.

Szlakiem pomników Kopernika.

MÓJ PRZYJACIEL PAWIAN

Odnalazłem fragmenty moich dawnych prób pisania dla dzieci, z circa 2004.

Wszystko zaczęło się od zeszłorocznego wyjazdu na działkę. Lało przez całe dwa tygodnie przez co Leon zostawał sam w domku letniskowym podczas gdy jego rodzice zbierali jabłka i śliwki w sadzie. Tata po każdym powrocie mówił że jeśli nie zebrałoby się tych owoców to przy tej pogodzie sad zamieniłby się w gigantyczną misę kompotu jabłkowo-śliwkowego. Leon na początku nie uważał że taka ogromna misa kompotu byłaby czymś złym, ale po dłuższym zastanowieniu stwierdził że nie ma wystarczającej ilości słoi i butelek aby ten kompot rozlać i w końcu że taką ilość kompotu mógłby chyba tylko wypić jakiś wielkolud albo smok. Niestety okolicy działki nie mieszkał ani żaden smok ani wielkolud, więc Leon stwierdził w końcu, że rzeczywiście rodzice mają rację zbierając opadłe owoce.

I właśnie tego trzeciego jakże nudnego poranka, podczas gdy Leon przeglądał po raz już chyba piąty jedyną książkę w domku, którą była „Historia Naturalna”, pojawił się ów gość. Był to nie mniej i nie więcej pawian we własnej osobie, a właściwie we własnej sierści. Siedział sobie jak gdyby nigdy nic w fotelu i łypał dyskretnie na chłopca swoimi małymi ślepiami. Pewnie myślał że nikt nie zauważył jego pojawienia się. Zuchwalec. Jak można tak po prostu pojawić się z nikąd bez zapowiedzenia? Leon nie omieszkał uświadomić tego pawianowi: „Trzeba było przynajmniej zadzwonić, jeśli nie na domowy telefon to na komórkę mamy, a jak już tak to przynajmniej powinno się powiedzieć dzień dobry.” Jak się okazało pawian był nie tylko impertynentem ale i niemową.

Mimo nienajlepszego zachowania Pawiana, Leon postanowił zainteresować go „Historią Naturalną”. Nietrudno się domyśleć że książka bardzo się gościowi spodobała; prawie przez cały czas przesiedział na oparciu fotela podskakując z wrażenia przy każdej kolejnej fotografii. Uśmiechał się dosłownie od ucha do ucha, i właśnie przy tych szerokich uśmiechach Leon zauważył, że jego gość zaniedbuje nie tylko maniery, ale i też higienę osobistą, a właściwie małpią.

Pawian zupełnie nie proszony wpakował się do samochodu, gdy nadszedł czas powrotu do domu. Leon nie protestował – przynajmniej nie było nudno. Ciekawe że rodzice w ogóle nie zwracali uwagi na dodatkowego pasażera; „To nawet lepiej” pomyślał Leon zadowolony z faktu że nie musi tłumaczyć rodzicom okoliczności pojawienia się pawiana.  C.D.N.

Postanowiłem poprosić Chat GTP 4 aby dopisał kolejna opowiastkę o Leonie i Pawianie. Wynik jest mniej satysfakcjonujący niż się spodziewałem, mimo tego że podobny eksperyment w przypadku opowiadania Immoral trip wypadł znacznie lepiej.

PROMPT: Mam pomysł na serię opowiastek o Leonie i jego wyobrażonym przyjacielu (pawianie). Napisałem już pierwszą, wstępna opowiastkę, gdzie poznajemy głównych bohaterów. Jest to pewnego rodzaju pilot. Chcę abyś napisał kolejną część -- czyli kolejna opowiastkę. Może tym razem rozgrywającą się w domu lub przedszkolu. Nie było to jeszcze powiedziane, ale Leon ma około pięciu lat. Oczywiście chciałbym aby było to zrobione w tym samym stylu i konwencji -- dziecinne przemyślenia i obserwacje Leona oraz ciekawe zachowania pawiana, no i aby była tej samej długości co pierwsza. Puść wodze wyobraźni!

Chat GTP 4:

Leon i pawian w przedszkolu

Leon uwielbiał chodzić do przedszkola. Lubił rysować, śpiewać piosenki i bawić się z rówieśnikami. Jednak pewnego ranka, gdy mama pomagała mu ubrać się i przygotować do wyjścia, Leon poczuł, że ten dzień będzie nieco inny. Zauważył bowiem, że jego wyobrażony przyjaciel, pawian, również wydaje się być w doskonałym nastroju i z niecierpliwością czeka, by pójść z nim do przedszkola.

Do przedszkola chodziło wiele dzieci i każde z nich miało własne małe zwariowane pomysły. Jednak pawian był czymś zupełnie wyjątkowym. Oczywiście tylko Leon go widział, ale to nie przeszkadzało mu w zabawie z innymi.

Kiedy nauczycielka poprosiła dzieci, aby narysowały swoich ulubionych przyjaciół, Leon postanowił namalować pawiana. Jego rysunek odznaczał się żywymi kolorami i dokładnymi szczegółami - od szerokiego uśmiechu pawiana po każdy włos na jego futrze. Kiedy przyszedł czas na prezentację rysunków, Leon z dumą opowiadał o swoim wyjątkowym przyjacielu. Dzieciaki były zafascynowane i chciały wiedzieć więcej.

Na przerwie w ogrodzie przedszkolnym, Leon opowiadał innym dzieciom o przygodach z pawianem. Jak przyszedł do niego na działkę, jak wspólnie przeglądali książkę i jak wsiadł do samochodu, wracając do domu. Wszyscy słuchali z otwartymi ustami.

Pawian, z zachwytem obserwując dzieci, postanowił zasugerować Leonowi nową zabawę. Wykorzystując gałęzie drzewa jako liany, podskakiwał z jednej na drugą, udając tarzającego się małpę w dżungli. Leon natychmiast dołączył do niego, a reszta dzieci śmiała się i klaskała w dłonie.

Po powrocie do domu, Leon miał mnóstwo historii do opowiedzenia rodzicom. O tym, jak wspaniale bawił się w przedszkolu z pawianem i jak wszyscy byli zafascynowani jego wyimaginowanym przyjacielem. A pawian? Siedział sobie spokojnie na fotelu w pokoju Leona, ciesząc się kolejnym dniem pełnym przygód.

Zachód słońca oznaczał koniec dnia, ale dla Leona i pawiana była to tylko kolejna strona w ich nieskończonej książce pełnej cudownych chwil. C.D.N.

Bracia

Bajeczka kórą wymyśliłem lata temu (w 2003!)---wersja tekstowa poniżej nie byłaby możliwa bez udziału i zdolności literackich przyjaciela, Jerzego Rybińskiego, alias 'Stryjaga'.

BRACIA

Pewnego razu, w pewnym miejscu świata,
Brat Duży przybywa do Wielkiego Brata
I prośbę przedstawia chyląc się mu do nóg
By ten w kwestii wzrostu Dużemu dopomógł
- zechciał bratu nadać kilkanaście cali…
by się podzielił…by ciut użyczył
Duży by tego bardzo sobie życzył.

Wielkiemu mimo zdziwienia koncept nagły się jawi,
Tedy tając obawę czy Duży to strawi rzecze:
- „Nie tylko wzrost mój jest wielki przecie…
Z miłości do was, co chcecie zrobię…
Zatem postawcie sobie na głowie
Ten dzban wysmukły…o tam stojący
Lecz uważajcie, aby niechcący nie zbić!”

„Dzięki niemu bowiem…od Dużych większy o łokieć,
z Wielkimi się zrównacie.
- Wniosek z tego drogi bracie,
bez ochyby się znajdziecie
w gronie Wielkich – w Wielkim Świecie.”

Duży Wielkiego dobrocią wzruszony,
z wdzięcznością pyta:
- Co ze swej strony mógłby dla brata czynić w odpłacie?
Wielki odpowie:
„Mój drogi bracie, ja bym zaledwie o drobiazg Cię prosił,
abyś codziennie mi wodę przynosił.”

Best of imperfect worlds.

Lev somewhat dissatisfied with his previous cosmic project, decided to embark on a more careful enterprise of world creation, and turn down the perfection parameter from maximum. This time he decided to play with the parameters of fundamental values, and observe how they'd influence the hedonistic dynamics. The idea was to run a simulation where hugs are set as having principal value, and consequently become the sought after currency -- in other words, hugs in that world were to be the sole wealth determinant. But in what sense 'hugs'? -- one may ask. Receiving them of course! And naturally they'd have to be of genuine sincerity; neither bought nor forced in any way. That is, hugs have value only if they're sincere and welcomed. But then how does one accumulate such wealth, given that hugs are such ephemeral phenomena? Surely, one can't be in possession of a great number of hugs. The only way one can accumulate wealth of this kind, that is, become a prosperous hugee, is to guarantee and maintain the existence of those willing to give those hugs, i.e. huggers (aka ready-to-hug beings). In other words, one can attain a high flux of hugs, and wealth would be interpreted as maintaining a high hug-flux. This can be done in many ways of course, and Lev calibrated the simulation with no limit on the degrees of freedom concerning valid hug-acquisition. Naturally those who attain the ability to reach and maintain a high hug-flux steady state, aka hugagogues  are sought after as raw models for guidance, whose wisdom would guarantee the attainment of wealth in that world. Lev has hypothesized that such a world would be among those that are the closest to being perfect without giving rise to any absurd consequences, which inevitably accompany perfection.

Explicit solution (formula) to the "truth table" recurrence relation.

If anyone has done any introductory logic, or has been introduced to representing sets in terms of their characteristic function, then the following binary matrix will look familiar. Each row is a distinct combination of the elements of some set. This is the matrix for a three element set. Call it matrix A.

1 1 1 ...
0 1 1
1 0 1
0 0 1
1 1 0
0 1 0
1 0 0
0 0 0
:
:

Some of the most common applications/interpretations of this matrix are:

i) For any natural number N (of columns), the rows of the matrix (of which there is 2^N) represent the set of all functions N→{0,1}., were N is a finite subset of the Natural numbers.

ii) The rows of a table thus generated, exhaust all truth value assignments to a propositional variable appearing in a formula of Propositional Calculus.

iii) Equivalently each row can be interpreted as the image of the characteristic function of a power set of some set S (we don't need to assume the axiom of countable choice since here we're dealing with finite sets). In other words each row of the matrix corresponds to a distinct combination of the elements of S. In fact the map f : ℘(S)→ ROWS (of the matrix) is a bijection: f(X)=r iff r(i)=1 iff i∈X, where we identify the matrix column indecies with the wellordering of the elements of S.

So clearly this matrix is a big deal, and subsequently the result here is an important one, since it basically compresses the entire matrix to a simple function of its rows and columns.

The way to generate the matrix in a way to ensure that all combinations are exhausted, is to follow the obvious pattern for column generation --- single iteration, double iteration, quadruple iteration, etc. In general, each column i has a 2^i-1  iteration of 1's and 0's (starting with 1's in each column).

Below is the proof for the explicit formula of such matrices recurrence relations, i.e. as a function of the matrix'  row and column number. That is, given only the row and column numbers, the formula gives the value that appears in the matrix on those 'coordinates'. That is, the formula takes the following functional form:

                                                                       θ : 2ⁿ×n → {0,1}

EXAMPLE: θ(k,i), where the row number is k = 7 and column number is i = 2, i.e. θ(7,2) = 0.

Now for the proof, which I have only given a sketch of in a previous post. Subsequently I lost the proof, and only had the formula, which bothered me, so I re-proved it last week. So here we go. Most of the pattern recognition which underlies the solution is obviously to be found in the matrix itself, and the diagram below intends to capture and make salient those patterns which may not be so obvious at first glance. The integer numbered columns, i.e. columns 1, 2, 3, 4,..., are the actual columns of the matrix A. 

To the right of each actual matrix column i, I list k(mod 2ⁱ), which is the first pattern that ought to be observed. Can you see how a single cycle of k(mod 2ⁱ) matches the length of the pair of iterated 1's and 0's, in column i? The next key observation, and perhaps the crucial one, is that the pattern of out actual matrix columns is basically a function of k(mod 2ⁱ). More precisely, note how each i'th actual column has the same pattern as k(mod 2ⁱ)−k(mod 2^i-1), with the exception of being slightly 'misaligned' and having some product of 2 in the place there 1's ought to be. I have indicated the k(mod 2ⁱ)−k(mod 2^i-1) rows with a yellow heading. Have a look for a while, and the pattern* should 'jump out'. Once you're convinced that it is so, all we need to do is (i) to rectify the 'misalignment', and (ii) the multiples of 2 integers instead of 1's. By noting that the 'misalignment' is also a function of column number, we see that adding 2^i-1−1 to k, does the trick. Finally divide all values in each i'th column by 2^i-1 and we're basically done. That is, our formula can be expressed as the function:

This formula is good enough and does the trick, but using the identity below, relating the mod and floor functions, it can make it more concise.

Giving the formula its final form. I guess it could be rendered even more concise and elegant with further algebraic fiddling, but I shall leave it there.

*Admittedly, a more rigorous proof is required than merely conjecturing that this pattern ought to hold in general. I may get around to it soon. But it does seem obvious that the proof is correct, since no surprises will arise in the relationship of the values of k(mod 2ⁱ) and k(mod 2^i-1) as both k and i increase.

If we switch the order of 1's and 0's in the table, i.e. if we let 0's precede 1's, like so (call this matrix B):

000...
100
010
110
001
101
011
111
:
:
Then the same reasoning yields a more elegant formula:

On the (nontrivial) non-uniqueness of the empty collection.

Consider a property P : "is green all over, and is not green all over".

Alice and Bob are friends. Bob reasons about the world, and accordingly conditions his understanding of what is or isn't logically possible to classical logic. In particular, not only there are no objects with property P, but there cannot be any such objects since P is a logically impossible property. Alice on the other hand, having travelled far and wide, and having seen curiosities such that no classically (as per classical logic) minded philosopher has even dreamed of, accepts paraconsistent logic as the correct way to reason about the world and its many wonders.

Alice and Bob agree that 'empty' (or 'the empty collection', denoted (e)) is an absence of any-thing, i.e. it is an absence of any object (or simply 'a set with no elements'). They also adopt a covention whereby saying that 'some-thing is contained by the empty collection' means the same as saying that 'that some-thing does not exist', e.g. 'four sided triangles are contained by the empty collection' is just another way of saying that 'four sided triangles don't exist'. This is the convention both Alice and Bob adopt, and agree on. Nota bene, impossibilia (impossible objects) are also contained by the empty collection, for since they can't exist, in particular they don't actually exist.

The duo however disagree about what counts as an object. Whereas Bob considers any object with property P as impossible, i.e. an object that cannot possibly exist, Alice doesn't. As a consequence, within their discussion about the world and its many wonders, the term 'empty collection' despite being correctly understood by both friends as having the same extension, is incorrectly assumed to have the same intension. The extension is the same since it is an absence of any object, but its intension is distinct, due to our protagonists' differences in reasoning about the world, which in turn bear on what counts as possible and what doesn't.

In particular, for Bob the empty collection has the property of containing all "objects" with property P, since that amounts to saying that such objects don't exist (since they can't exist!). Alice would disagree with Bob with regards to such a property --- she would negate it having such a property outright, by saying that it's not the case that the empty collection contains all objects with property P, since (after all) the existence of such objects is consistent with her view of the world. So if she were to accept that the empty collection had such a property, it would amount to saying that the empty collection is not empty after all. Instead, Alice asserts the negation, i.e. that there are some objects with property P that are not contained by the empty collection.

What does this mean? That Alice and Bob aren't talking about a unique empty collection, but rather two distinct ones. To eliminate confusion they decide to introduce yet another convention whereby they index those distinct empty collections by what makes them distinct --- in this case the distinction is conditioned on who is considering the empty collection, i.e. (e)-A, and (e)-B. But since the differences in the intension of those two terms are in virtue of the reasoning system (logic) adopted by either friend, we can conclude that the intension of "empty collection" is logic relative.

Unlike the act of conditioning the meaning of 'the empty collection' to contexts which deliver contingent distinctions, e.g. what a caveman, a poet, a chemist or a physicist would consider as 'empty', the difference in the meaning of "the empty collection" in the case of Alice and Bob arises out of how such concepts can be reasoned about in principle. As such it is a distinction in the meaning of 'the empty collection' in principle, i.e. in principle that concept has no unique meaning.

Also this is a nontrivial claim, since it's possible (conceivable) that Alice and Bob never met, and so never discovered the discrepancy in the meaning of 'the empty collection'. In other words, it's a nontrivial result since it's conceivable that there could be rational agents confined to a single reasoning system only, thereby not being capable (lacking the necessary epistemic condition, which is the act of abstracting away from the preferred reasoning system) of seeing the fundamental non-uniqueness of the notion of 'the empty collection'.

To sum up: Alice's notion of 'the empty collection' is not the same as Bob's notion of it is. Concisely speaking (e)-A is not identical to (e)-B, which yields the truth of the claim '(e)-LP is not identical to (e)-classical'.

But let's assume per impossible that (e)-LP is identical to (e)-classical; hence x is in (e)-LP iff x is in (e)-classical. Also pick the object a (such that Pa) as an LP-possible object (a is an LP possibilia). This is a fair assumption since not all contradictory properties need be impossible in LP, thus allowing objects with such properties to be legitimate possibilia. Hence a is not in (e)-LP, according to the convention adopted by Alice and Bob. But according to the same convention, all x such that Px are elements of (e)-classical, i.e. all x such that Px are classically impossible (are classical impossibilia). In particular a is in (e)-classical, but that means that a is in (e)-LP given the hypothesis (identity assumption), but we assumed that a is not in (e)-LP, which yields a contradiction.

Therefore, adopting classical logic as the one governing this proof (the metalogic here), and reductio ad absurdum as a valid proof method, it follows that (e)-LP is not identical to (e)-classical, as required.

The key idea of the above discussion can be compressed to saying that although the extension of classically impossible objects can be expressed as the extension of the set (CI) of objects that satisfy some inconsistent property, since those two are necessarily coextensive in classical logic, but the extension of paraconsistently impossible objects cannot be expressed as the extension of CI. Hence, the intension of 'no possible objects' is logic relative (obviously?).

Note: 'element of' and 'belongs to' and 'is in' are terms expressing binary relations that I tentatively use to denote some kind of association of an impossibilia with the empty collection, or the empty set (or empty sets, since I'm arguing that they're needn't be unique across distinct theories). Note also that for each single theory the empty set still remains a unique object/element, as theories are usually based on a single logic. (This assumpton needn't be correct.)

One may rightfully object, and some have, that there's a fundamental problem with asserting that anything is an element of the empty collection. But suppose I claimed that the objects it 'contains' are non-existent, but by doing so were I to wish to retain the meaningfulness of such 'objects' I would be committing to some kind of non-eistic ontology, which posits non-existent objects. But those non-existent objects are objects of some kind after all, and thus cannot be said to be elements of the empty collection, which contains no objects whatsoever. So suppose I reject non-eism, and say that when I say 'a is an element of the empty collection', in no way do I wish to commit to a's existence --- not even as a non-existent object --- i.e. there is no a. But then what is it that I am talking about when I refer to a? I'm not refering to anything apparently, by definition.

I choose the second route, i.e. the one that doesn't commit to an non-eistic ontology. 

(To be continued.)

How empty is { }? Different orders of the Empty Set. Coursera Forum discussion.

Here is a raw cut-and-paste of the exchange on this topic from a Coursera forum.

I'd like to thank those who provided constructive criticism, or just picked at the idea with an unbiased razor of skepticism. Apparently most of the interlocutors were inspired, and some have even admitted to having been led to think in a new way --- to a philosopher, there's no greater reward :)

It's this sort of exchange that allows the idea to mature, and hopefully catch on :)
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Mariusz Popieluch· 2 months ago
Let me introduce the discussion with a little humor.

The Poet, the Chemist, and the Physicist.
The trio met one day over some wine and discussed matters of rhyme, thyme and time. Soon the conversation turned to the notions of nothingness and emptiness - what is empty? The poet finished his glass, and pointed to it -- as far as I am concerned this glass is empty, as it is devoid of wine - the sparkling grape, the drink of gods. Hold on a minute! -- exclaimed the chemist -- surely it's not empty as it contains air - we'd have to pump all the air out of the glass, creating a vacuum in it, and then and only then would it be empty. It wouldn't make any difference to me -- replied the poet, shrugging his shoulders. Please let me interject at this juncture dear fellows -- interjected the physicist -- and let me put an end to your obvious confusion. Vacuum, as you describe it doesn't cut it at all, since we know that even empty space is a breeding ground for virtual particles whose immediate annihilation results in what we observe and call vacuum energy. The poet looked at the physicist with a frown of suspicion -- I know nothing when I see it good man, and I won't let anyone tell me otherwise - let the bartender settle this matter - haloo, good fellow! Another round please!

-------

The above humorous scenario pertains to natural language and some theories of varying degree of conceptual refinement, but the idea extends nicely (I believe) to formal languages, theories and logics too, thus generalizing the idea of emptiness. In particular it intends to do to the notion of emptiness and it's various formalizations such as the set theoretic ∅, what Cantor has done with the notion of infinity - show that there is structure within it, and varying orders of it.
I know this is a radical idea, but rest assured that the idea of varying orders of infinity wasn't initially taken lightly either - even today many still think it's preposterous. But interestingly enough, there seems to be a demand for such a hierarchy - hinted at in the literature - but I'm not aware of a formal theory of various orders of the empty set being developed.
The general claim is that what counts as empty is dependent on the strength of a logic underlying the theory in which it is defined and the expressibility (richness) of the language of the theory. That is, in some theories there are more objects that fail to be members of the empty set (e.g. to the physicist there is more to the contents of the glass than wine and air), thus rendering the empty set somehow emptier (to put it simply).
Or to put it another way - we could just define the order of the empty set by the cardinality of all the impossible objects in a given language/logic/theory. Those impossible objects are not actually members of the empty set, but rather are assigned to it in some sense - by saying that some object (this includes linguistic objects) is an element of the empty set (in a given theory), is just another way of saying that this object is impossible (in that theory).
Example.
Consider classical logic CL and some paraconsistent logic, say LP. In LP contradictions are not considered as impossible. Now define a property G: of being green all over, and not being green all over. Now, in CL any object with that property is considered as an impossible object, take the object a, so in CL we obviously have Ga ∈∅, but in LP we have Ga ∉∅. Hence the suggestion to index ∅ by logics - this may become crucial when talking of various theories in some metalanguage/meta theory.

Philosophically speaking "{ }" is an object with a fixed extension across theories (it has no elements), yet varying intension. What the above aims to show is that the varying intension can be accompanied with a varying quasi-extension, i.e. rarity as opposed to cardinality.

Mariusz Popieluch· 2 months ago

Consider two urns in a marble shop. The shopkeeper labelled them in the following way - urn 1 holds only (and can only hold) white marbles, whereas urn 2 can hold both white and black marbles.

Now consider a new assistant that has just been offered an apprenticeship in the shop of his dreams (he loves playing marbles), and is not as yet familiar with the arcana of the labeling system, i.e. he hasn't as yet been told by the shopkeeper what the restrictions on the contents of the urns are.

Now consider the statements 'urn 1 contains no white marbles' and 'urn 2 contains no white marbles' - they entail different things, depending who is exposed to that information - the shopkeeper or the assistant. In particular, to the shopkeeper it means that 'urn 1 is empty'.

On a more technical side, we can substitute the terms 'has no white marbles' and 'is empty' interchangeably salva veritate in the context of urn 1 (charity permitted, for naturally one could generate oddly sounding sentences). This cannot however be done in the context of urn 2.

David M. Kaziska· 2 months ago
In your first post the set, {wine in the glass}, is empty after the wine is consumed (neglecting any residual wine).  For the poet's own purposes it is empty but we may his or her statement that it is empty as colloquial in light of the chemist's and physicist's later statements.  I'm not seeing mathematical consequences which would require redefining the notion empty sets.  {wine in the glass} may be an empty set but {air in the glass} may be nonempty.  

In your later post, the shopkeeper reasons as follows.  (P1)  All marbles in Urn 1 are white, (P2)  There are no white marbles in Urn 1, therefore (C)  Urn 1 is empty.  The assistant has (P2) but not (P1) and cannot draw the conclusion.  Urn 1 is still empty, though, it's not a matter which is relative to the observer.

Mariusz Popieluch· 2 months ago
Thank you for your reply David. I'll do my best to address your observations, and further clarify the idea. :)

The later post intends to illustrate the idea that in some theory T based on some logic L1 some formula F may express an impossibility (which we can express as that formula being "an element" of, or assigned to, the empty set), whereas the exactly same formula in an analogue to T based on a logic L2 which is stronger than L1 may not come out as impossible, and as such cannot be expressed as "belonging"/being assigned to the empty set.

Note: "element of" and "belonging" are terms I tentatively use to denote some kind of assignment of F to the empty set (or empty sets, since I'm arguing that they're needn't be unique across analogous theories based on distinct logics). Note also that for each single theory the empty set still remains a unique object/element, as theories are usually based on a single logic.

The epistemic states of the protagonists of the marble shop illustration (informal context) correspond to the assumptions/axioms/conditions/semantics of theories based on distinct logics in the formal context.

Some logics distinguish semantically the propositions "p & ~p" and "p & q". In particular they consider the first proposition to take a fixed value for all valuations (contradictions are always false, and only false in classical logic). Some other logics do not make that distinction.

Likewise the shopkeeper distinguishes the two propositions concerning the urn contents, whereas the assistant doesn't. Try to think of the urn labelling system and the protagonists' distinct epistemic states as an analogy for the distinct meta-linguistic propoerties of analogous theories based on distinct logics, and the propositions concerning the urn content as object language entities, i.e. formulae.

But if the informal analogies muddle, rather than clarify the idea, I'll be happy to talk more formally about it. :)


Louise Craven· 2 months ago
Every set is a subset of the relevant domain of discourse. Thus if we are only talking of natural numbers, the empty set is the set containing no natural numbers. The poet was thinking of drinks when he said his glass was empty, so it truly was. The chemist and physicist tried to confuse him by exploiting the fact that he had not explicitly stated the domain of discourse, and they were also right about the nature of the empty set within their individual domains of discourse.  In the marble example, the difference between the shopkeeper and the assistant is not in the 'size' of the empty set, but in the inferences they can draw from the fact of emptiness, which depends on other information.

I am, however, intrigued by your suggestion and it may be that in some applications of mathematics to real-world problems, the size of the empty set in the model may need to vary according to the significance of 'nothing' in the area modelled. For instance, complete extinction of a species is of much greater significance than the remaining existence of a number of breeding pairs much smaller than the population at a previous time.

I shall go now and give it more thought....

Mariusz Popieluch· 2 months ago
Hi Louise - thank you for taking the time to read the OP.

The informal anecdotes are mere illustrations of the general idea.

In the first scenario I intended to show, via the relevant domains of discourse illustration,that given the same language (in this case natural language), but distinct theories, identical propositions (formulae, i.e. linguistic objects) entail different things - in particular, where in one theory "no wine in glass" does entail "glass is empty", in other theories it doesn't. I stress again, that this is merely an illustration of meta theoretic properties, rather than a discussion concerning restricted quantification to relevant domains of discourse per se.

Think of the Poet, the Chemist and the Physicist as analogues of theories of "The stuff on the table, at the local restaurant.", albeit based on dictinct logics. In the poet's logic "there's no wine in the glass" can be associated with the empty set - this however isn't true in the chemist's or physicist's logics. Likewise in the chemist's logic "there is no liquid and no gas in the glass" can be associated with the empty set - but again, this isn't true on the physicist's logic.

As for the second scenario, please see my above reply to David. And please do not shy away from addressing the formal content of the OP.

Andrew Kelley· 2 months ago
First let me say that I enjoyed reading this post. It made me think in a way I hadn't thought before.

I think you may be on to something with recognizing different orders of the empty set. However I am as of yet unable to think of any examples where the knowledge of such different orders is relevant to solving a problem. Are you?

Mariusz Popieluch· 2 months ago
It may serve as the ontological foundation for a theory of comparative impossibility.

Francisco Vasconcelos· 2 months ago
I think that the analogy between empty set and infinite set is not the most correct one. The evident oppositions would be between empty and complete set, and infinite and infinitesimal amounts.

So on the empty/complete side, I'm not aware that there are different orders of completeness. Both these terms are usually used as binary categories applied to other sets, both finite and infinite. Yes, their meaning varies with language context, but so do everything else, such as the number "2", the operator "+", or the word "number".

On the other hand, I think it's easier to make the claim that there are different orders of infinitesimal quantities. For example, think about the probability of picking the number "5" at random from the set of all natural numbers. Since the set is infinite, the probability is 0. However, it seems that this "0" is still bigger than the "0" probability of random sampling the number "5" from the set of all real numbers, since the pooling set is infinite to a higher degree than the previous one.

Andrew Kelley· 2 months ago
I'm not buying that you can say that one 0 is bigger than another 0. Zero is zero. Can you prove it?

Mariusz Popieluch· 2 months ago
Francisco, thank you for your reply -- you're right that it's not the most correct analogy, as it's not intended to be a directanalogy, i.e. a dual of sorts. It's a weak kind of analogy, in the sense that in both cases of infinity and nothingness/emptiness we're dealing with concept which apparently don't admit to degrees. Whereas Cantor developed the idea of bijection as the criterion for equinumerosity, I'm using it in developing the notion of the rarity of some empty set, which is currently tentatively defined to be the cardinality of the set of all formulae which express an impossibility (aka impossibilia) in a given logic L, thus yielding an indexed empty set ∅_L.

As I said before, in the above response to Louise, the contexts of relevant discourse analogy is again merely an illustrationof the more precise idea. See the above reply.

The observation you expressed in your last paragraph puts you in good company. In the context of probability theory the idea of different orders of “zero” has been hinted at by Andrey Kolmogorov and Bruno de Finetti as a possible candidate to solving some probability theory paradoxes – “Like Kolmogorov, de Finetti is occupied mostly with probabilities defined directly on arbitrary uncountable sets; but he views additivity differently, and is led to such anomalies as an unlimited sequence of layers, like an onion, or different orders of zero probabilities that add up to one, etc. ” (E.T. James: Probability Theory, the Logic of Science, 2011, p.656).

Francisco Vasconcelos· 2 months ago
Andrew, try not to read my "0" as the integer number 0, but as an infinitesimal quantity that is infinitely close to zero, something like the result of limx→∞1x. As infinity can have different sizes, the result of this limit quotient should be able to have different sizes too.

Mariusz, thanks for the info

" different orders of zero probabilities that add up to one"

this is definitely interesting and worth checking out.

Regarding your emptiness orders:

"I'm using it in developing the notion of the rarity of some empty set, which is currently tentatively defined to be the cardinality of the set of all formulae which express an impossibility (aka impossibilia) in a given logic L, thus yielding an indexed empty set ∅_L."

I get it now. So I guess you're trying to say that in different theories the empty set can be connected to a set of propositions that can have a different finite or infinite size.
Another question: hypothetically speaking, do you think that within a single theory L (maybe with number references), it is possible to represent empty sets with different sizes in this sense?

Andrew Kelley· 2 months ago
We can mathematically prove that infinity can have different sizes by using bijections. But we cannot mathematically prove that limit of 1/x as x approaches infinity is not equal to the integer zero. In fact, we *can* prove that, for example, 0.999999 repeating is exactly equal to 1. Not a number which has an infinitesimally small difference than 1. 1.

Francisco Vasconcelos· 2 months ago
Andrew I'm still trying to figure out what different kind of "0" can mean, it seems that this concept raises some issues in probability theory, but lets forget that for a moment and define the following:

For any functions f(x) and g(y), assume that
limx→∞f(x)=0
limy→∞g(y)=0

Now suppose the following hypothesis:
For any arbitrary ϵ, there is NOT a bijection between all possible values of g(y) and f(x), for x,y≥ϵ

If this proposition is true, then the limit of g(y) and f(x) being "0" might have different meanings, since we know that for any given ϵ, one of the sets will always be larger than the other.


Mariusz Popieluch· 2 months ago
@Francisco: "Another question: hypothetically speaking, do you think that within a single theory L (maybe with number references), it is possible to represent empty sets with different sizes in this sense?"

Well, personally I'm not entirely on-board with that idea, for the reasons I stated in my reply to David above: "Note also that for each single theory, the empty set still remains a unique object/element, as theories are usually based on a single logic."

My position is such, due to what I mean by "different orders of emptiness". But this is not to say that I'm not open to the variation of this idea, which you and the company of famous thinkers suggest. :)

Andrew Kelley· 2 months ago
Francisco, thank you for this example. I think you may be right but I am finding it extremely hard to wrap my brain around it.

Hayden VanIderstine· 2 months ago
Francisco, in order for the hypothesis that there exists no bijection between all possible values of g(y) and f(x) when x,y>ϵ, then it is necessary that the cardinality of the set of all possible g(y) with y>ϵ be different from the set of all possible f(x) with x>ϵ. For this to be the case, then both sets cannot both be of any of the following cardinalities:

Finite
Countably infinite (set of naturals)
Uncountably infinite 1 (set of reals)
Uncountable infinite 2 (set of all functions defined from the set of reals to the set of reals)

What I am wondering, is what is the domain and range of f and g, because if they both have a range being a subset of the real numbers (or the entire set of real numbers), then there necessarily exists a bijection between them.

Francisco Vasconcelos· 2 months ago
Hayden,

I'm not sure I follow you. The way I see it, for there to be no bijection it is only required that the domain of f(x),x>ϵ has a different cardinality of g(y),y>ϵ. So, for example if the domain of f(x) is a subset of the rational numbers, and the domain of g(y) is an interval of the real numbers, then there is no bijection.


Hayden VanIderstine· 2 months ago
In my opinion, excellent post Mariusz. I am thinking in a new way now thanks to you.

What we consider as empty is dependent on what we acknowledge exists.

{n∈N | 1 < n < 2} =  ∅
Exactly  |N| elements fail to be in the set {n∈N | 1 < n < 2}

But exactly |R| elements fail to be in the set {x∈R | 1 < x < 2}.

According to your definition of the rarity of some empty set, would the rarity of the empty set in a logic which isn't strong enough to construct the reals, but is strong enough to construct the naturals, be |N|, because there are only a countably infinite number of sentences like 1 < n < 2 which could be made which are impossible?

(btw, I'm not well versed in formal logic, nor am I sure that there exists a logic which is strong enough for the naturals, but not enough for the reals).

Mariusz Popieluch· 2 months ago
Hayden - thank you for your insightful reply and interpretation of the ideas in this thread. Your observations, and question also made me think more carefully about what I'm proposing.

The natural numbers, or Peano arithmetic, is a first order theory, whereas the it is not possible to characterize the reals with first-order logic alone since the supremum axiom of the reals quantifies over subsets of the naturals, and is therefore a second-order logical statement. See the list of axioms here: https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach

The stronger logic in which the axioms of R are expressed is second order logic, which is stronger than first order. So to answer your question directly - first order logic is the logic "which is strong enough for the naturals, but not enough for the reals", for the reasons given above.

(Note on the terminology: Logic B is stronger than logic A iff all theorems of A are B therorems, and there exists some B theorem that's not an A theorem. Or equivalently B is a stronger logic than A iff the set of A theorems is a proper subset of B theorems.)

Mariusz Popieluch· 2 months ago
Also, to clarify - "What we consider as empty is dependent on what we acknowledge exists." is not entirely correct.

Rather "What we consider as empty is dependent on what we acknowledge can exist.".

Pedro Forquesato· 2 months ago
Hello Mariusz,

Nice post! It brought a good discussion, and that is the purpose of this forum. I think your stories are good examples of the importance of clear and rigorous (for example mathematical) sentences in philosophy (and other studies), and the danger of "spoke communication".

Let's try to solve it by translating to "mathematical language" what they are saying. In the case of the urns, for example, when the apprentice says "the urn is empty", he means (lets define U as urn and W as the set of white marbles):
U = { }
While when the master says "the urn is empty", rigorously he means:
U intersection W = { }

So while in English what they say is the same, actually they are saying different logical propositions, and thus it is not paradoxical that the second might be true while the first is not. It is not the empty sets that are different, it is the translation from English to logic that differs. (Naturally the same argument is valid for the poet's glass).

Mariusz Popieluch· 2 months ago
Thank you for your reply Pedro - if you're interested in a mathematical treatment of the idea, don't look at the intuitive illustrations of the marble shop, and the trio drinking wine. Instead focus on the content of the OP that follows the phrase "The general claim is...". :)

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To be continued...

What are dreams, mum?

I was a very inquisitive child, or at least so I've been told, and the beginning of my ability to speak was marked with an onset of a daily torrent of interrogations, which I subjected my mother to. I used to ask my mum -- the all knowing oracle -- about basically anything and everything.

"What are dreams, mum?" -- I inquired one day. This question, as she later told me, had stumped her at the time -- she didn't expect it coming from a six year old, any more than she knew how to explain it to one. 

"Well" -- stalled the oracle a little as she gathered her thoughts -- "they're a kind of recapitulation, night-time thoughts if you will, a replay of the experiences you had during the day -- that includes the people you may have encountered during that day, such as your friends and your teachers". 

"So if I were to encounter a monster" -- I pried (I was obsessed with dinosaurs and monsters) -- "I may dream about one the following night?"

"Yes, although as you know there really are no monsters" -- my mum took the opportunity to reassure me.

"So if dreams are night thoughts about the things I encounter during the day" -- I persisted -- "why do I sometimes dream about monsters?"

Here, my mum was a little stumped, and didn't know what else to say -- after all she didn't have a degree in psychology. After a little while she refined her answer -- she always came through, being the oracle -- "You may also dream about any thoughts that you may have during the day -- why, you may even dream about your auntie who lives in America!"

"What if I think hard enough about something I like" -- the merciless torrent continued -- "I may dream about it too?" -- I asked, brimming with excitement about having devised an idea for an experiment to be conducted over the coming days and nights.

Now, having looked back, I realize that I've never actually stopped asking such questions.

Friendship

One of the most beautiful qualities of true friendship is to understand and to be understood.
Lucius Annaeus Seneca

It's difficult to overstate the importance of this.

Is necessary absence the same as impossible presence?

?