I thought about you before I knew you!

Having a somewhat romantic disposition, I used to indulge in the following fantasy, in my former, bachelor years, when my passion for philosophy and science left little room for a long term relationship: 

"Suppose I fall in love sometime in the future, and meet the girl of my dreams. If that indeed happens, then she is alive today somewhere, going about her everyday life, oblivious to the fact that we're going to be together, and the fact that I'm thinking about her at this very moment." I indulged in the fact that I could think of her before we even met. 

But was I really thinking of her, or merely about a set of people who merely satisfied the predicate 'my next girlfriend'? This is analogous to thinking about the winner of a race before it has started. But it was difficult for me to accept that I'm merely thinking about a potential person, given that she already existed. Surely it wasn't determined who she is (going to be), before we actually met, and therefore at the time of my romantic deliberations. But such speculations aside, surely the predicate picks out a single person, although it can be said to apply to a number of potential candidates, as it were. At least the romantic mind tends to folly in this manner (I am thinking about her) where cold reason suggests otherwise (I am thinking about a set of potential the ones).

However, presently when I say to her "I thought about you before we met" it seems to be true, because now we know that it would actually be her. This is truly romantic! If so, then we have logically validated, or at least salvaged some of the reality underlying my spells of romantic imagination! And at the same time, it is also true that in some sense I was thinking about her before I met her.

LETHAL SPELL ABILITY thought experiment.

I had thought of this interesting hypothetical scenario recently, which could easily be extended to a thought experiment. Imagine that each person, over the age of, say, 16 was given the magical ability of having another person drop dead by thought alone, e.g. 'Adolf!', and Adolf drops dead. The ability would entail perfect concealment of he act, i.e. it would be impossible to know who had cast the deadly spell. This part is important, so those exercising this magical power would do so with full knowledge of its effectiveness, and impunity. Moreover, I believe that to many, only the presence of such anonymity would be a necessary condition for casting the spell at all.

This is interesting for a number of reasons. I wonder how many people would go ahead and actually use the spell? Also, suppose the version of the magic ability such that the total number of spells was unlimited, with perhaps some daily limit. And that each spell would have to be cast at an individual, e.g. general spells like "may all people above 6 feet drop dead" would be disallowed, which doesn't mean that one couldn't find out who the people like this are and cast the spell on them. At what rate and in what sorts of patterns would people start dropping dead? 

I have been thinking about this, and wondering what kind of people would jump at the opportunity of exercising such power.

For example, suppose that in the original scenario (of having a single spell only), we let the time to cast the spell last for a month. The point being that existing grudges, prejudices, and feelings of hate would be sorted out right away, so to speak, instead of having the scenario allow to wait for a time when an otherwise unwilling to use the spell person would be forced to do so in self defense. 
So suppose a month goes by and all the spells had been cast. It follows that we're left with a population that is at most smaller than before by the whole number of people (magicians) who were endowed with the ability to cast the spell (the case when no magicians are killed before casting their own spell), and roughly (±1) at least smaller than before by half the number of magicians (the case when half the magicians are killed before casting their spell).

Now imagine 2 alternative outcome-worlds:
--world 1 consisting of all the spell casters and the remaining youth under 16. So this world is left with those that though it to be better to cleanse it one way or another. And those who got eliminated no longer exist.
--world 2 consisting of all the spell victims, and all of the under 16 that were present before any spell was cast, i.e. the original, under 16 population. So basically this world consists of everyone minus the spell-casters.

I wonder how those worlds would evolve in time. World 1 has less people remaining, but let's suppose the populations to be sufficiently large, as not to be in danger of extinction after the cleansing process. I wonder how those worlds would differ, given what kind of people occupy it.

I also wonder if people would just annihilate themselves into extinction, on the version of the scenario where there would be no limit on the number of lethal spells. It would be like giving anyone access today to the current nuclear arsenal deployment.

I'm sure there are other aspects of this thought experiment that I haven't raised, and which could be quite interesting. I'm thinking of writing a short story based on this scenario as the underlying premise. No prizes for guessing that Borges and Lem are among my inspirations! 

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

A paraconsistent emoji.

Whose smile is both dead and alive at the same time.

#NotAll: a game.

This is an early draft, so it contains some repetition, and is bound to contain some omissions and even minor errors. I'd love to hear what people think. Any feedback, suggestions and criticisms are most welcome! :)

GAME IDEA
This is just a very early, rough draft. It is initially intended to be a physical board game like Scrabble or Monopoly or Chess, but I see no reason why it couldn't be simulated as an application. 
The game is a bit like logical scrabble, with the key difference being that you're describing an object (a toy possible world) rather than just concatenating letters to create words. So the sentences you formulate in this game are meaningful in a way that words are not in Scrabble. It would be no trickier than Scrabble in one sense and chess in another sense. 
The idea of the game is to make true, first order logic sentences about toy possible worlds, which consist of and only of different species of animals, incorporating elements of both deductive and inductive reasoning. So the purpose of the game is to have both educational and entertaining value.

MOTIVATION
Speaking well is a knack. Speaking well and truly is a virtue.
The game could be for those that teach introductory logic, or any other subject for that matter, who would play with their students---the game should be graspable to 10 year olds, and appealing to nerds and smart kids in general. So all people who like puzzle games would benefit from its entertainment value, and teacher-student combos would benefit from its educational value. And not only the educational value of sharpening their linguistic and reasoning skills, but also reinforcing knowledge about the hierarchical relationships present in any subject matter, e.g. take the Linnean taxonomy, geometrical figures, numbers, or even philosophy! 

LANGUAGE
- The language is mondaic first order logic. Moreover it's simplified to consist of atomic formulas only (and their negations), so to that end we employ bounded quantification. To make things easier no other logical connectives aside from negation are used. In addition, each animal comes in various colors (say, green, pink, blue), and can be either happy or sad (in future versions of the game I foresee animals that can be both or neither).
- So the kind of sentences that are made would be for instance 'All giraffes are happy', 'Not all green giraffes are sad', 'Some snakes are red', 'Some happy snakes are blue', 'All happy pink animals are horses', etc. (We could also partition the species of animals into classes, like mammals, reptiles, birds, fish, and add the corresponding predicate letters to the language.)
- The bounded quantification approach works in a way to dispose of the need of any other connectives except negation. 

Remember that the universe is 'Animals', so instead of saying 'For all Animals x, if x is Green then x is Happy', we say 'For all Green Animals x, x is Happy'. This way we have replaced the need to use the material conditional connective, by shifting the role of the antecedent to that of restricting the range of the universal quantifier 'All Animals'. 

Similarly with existential quantification instead of saying 'There exist Animals x, such that x is Blue and x is Sad', we say ''There exist Blue Animals x, such that x is Sad'. Similarly, as in the previous example we have replaced the need to use a connective corresponding to conjunction, by shifting the role of one of the conjuncts to that of restricting the range of the universal quantifier 'All Animals'. 

STARTING OFF
- Players have an unlimited number of universal quantifiers to start off with.
- Predicate letters (species, animal classes, colors, attitude), negations and existential quantifiers are drawn at random from a box.
- Animals are drawn at random from a box.
- The game starts off with a handful of animals, say 5, picked at random, and placed randomly on a 8x8 grid, say, which constitutes the initial state of the world.

RULES & TURNS
- Quantification over empty domains is not allowed. Consequently sentences that quantify over an empty domain (in a given turn) receive no points.
- After the world has been created, players draw at random from a box of predicate letters, existential quantifiers and negations, and subsequently aim to formulate true sentences about the world. That's how points are earned. Obviously drawing a negation would be a little like drawing out an existential quantifier, since we have unlimited universal quantifiers, so saying 'Some giraffes are sad' and 'Not all giraffes are happy' are both equivalent, but whether the existential quantifier or the negation will be useful will depend on the predicate letters (happy, sad) that we have already drawn out (and hence at our disposal), and the state of the world.
- Once players have picked some predicate letters they can either construct a true sentence, if possible, or draw from the box of predicate letters,negations, and existential quantifiers again, or draw from the box of animals to add another animal to the world.
- The reason why existential quantifiers are scarce is because true existential sentences cannot be falsified, unless we introduce the dying off of random species or individuals (or any class, i.e. species, animal class, color, attitude) every couple of turns.

- Obviously true universal sentences at some stage of the world (any given turn) may be falsified, whereby someone draws out and adds an animal to the world that is a counterexample to that sentence. This is the key, inductive element in the game.
- So it would be a good idea to strategize probabilistically with the universal claims, by hedging one's bets via varying the restrictions of the universal quantifiers. That is, it's easier to falsify 'All crows are blue' than 'All happy crows are blue', since any (happy or sad) non-blue crow falsifies the former sentence, but only a happy non-blue crow falsifies the latter. But then scoring should reward strong hypotheses, i.e. more easily falsifiable hypotheses. I turn to that matter in the next section.
-Once sentences become falsified, their components (i.e. predicate letters quantifiers and negations) can be used to formulate another sentence.
-In any given turn a player can dismantle their true universal sentence X in order to use its components to formulate another sentence Y. In that turn X shall cease to receive points and Y shall receive (or starts receiving) points instead. One may want to do this if they have gathered sufficient predicates to formulate a universal sentence that has a higher score.
-Similarly, in any given turn a player can dismantle their true existential sentence X in order to use its components to formulate another sentence Y. Then X contributes zero to the final score, unless we introduce random dying off of individuals.

POINT SYSTEM & SCORES
There is a set number of turns, and the objective of the game is to end up with the highest amount of points after the last turn.
- For example we may set say 100 possible individuals, e.g. 10 species, 5 colors, 2 attitudes. 
- The game starts with 5 randomly picked individuals.
- Existential sentences are worth less then universal sentences and get a once-only score (unless we introduce dying off of individuals), whereas universal sentences accumulate points, each turn, for as long as their truth "survives". So the final score that an universal sentence gets at the end of the game is how much it is worth, multiplied by the number of turns that it stays true. The score of universal sentences will be proportional to their quantifier's range, or more precisely, to the probability of the sentence's counter-example. Let me explain the rationale behind this in the following example.

EXAMPLE: Consider a world consisting of animals, of which half are sad and half happy, and each animal is either blue or red with equal probability. We can represent the individuals occupying that world by the set {B, R} x {H, S}, e.g. (B,H) corresponding to a Blue and Happy Animal. Now consider the sentences 'All Animals are Happy' and 'All Blue Animals are Happy'. The probability of a counterexample to the first sentence is P(~H|A) = 1/2, because there are as many happy as unhappy (sad) animals. Similarly, the probability of a counterexample to the second sentence is P(~H|A & B) = 1/4, because half of all animals are blue, so not just any sad animal would suffice for a counterexample here (as it did for the first sentence) but one that on top of that is blue. We could transform those probabilities to some integer values for the actual score, so perhaps have P(X)=1/n give a score of  |A|/n, where |A| is the total number of animals.

- Existential claims would be a function of their strength, since you need more predicate letters (remember you only are allowed to draw one per turn) to create a stronger one, e.g. 'Some cows are happy' would be less valuable than 'Some blue cows are happy'. I'm inclined to set the score for existential sentences to be inversely proportional to the probability of their truth. Here's the rationale. It's not only easier to make weaker true existential claims, because they're more likely to be true, but also formulating stronger claims requires more predicate letters, and you only get one per turn. So following the world from the above example, and sentences 'Some animals are happy' and 'Some Blue Animals are Happy', we see that the probability that the first one is true is P(H) = 1/2 (so the score would be, say 2) and the probability that the second one is true is P(H & B) = 1/4 (so its score would be, say 4).

A scoring system as described above would introduce an incentive to strategizing, but to be honest I'd prefer a game that focuses on the linguistic and logical aspects, rather than the quantitative ones that involve sophisticated probabilistic strategizing that gives the players an edge. If the game was an application, then the scoring could be done automatically, so players would just follow their natural (and naturally flawed :) intuitions, making the game that much more fun! :)

APPLICATION VS PHYSICAL BOARD GAME
- I had this idea, if the game was an application, the app would prompt you about the aletheic state of your sentences, e.g. to have true sentences light up green. Similarly, have the app prompt you whenever one of your sentences becomes falsified by a change in the state of the world (i.e. addition of counterexample animal, or random dying off of an animal) by having the sentence light up as red. This would be very helpful, so you don't have to keep track of the aletheic state of all your sentences. You could then re-use the elements that make up that false sentence to construct another one. Also if the game was to be an application, all the scoring would be done automatically.
- The random dying off of individuals could be made by having the world as a grid, say 6x6 or 8x8 or 12x12 and you'd throw a pair of dice for each coordinate to determine the individual that dies. Of course all of such physical limitations (Platonic solids as dice, or other contraptions) would be easily surmounted if the game was an application.
- If created as an app the toy possible worlds could be customized by potential players themselves to fit any subject matter they like; not merely animals, but fantasy characters, aliens, numbers, geometrical shapes etc. as well.

DESIGN
I'm currently making some mark-up "screenshot" sequences of turn-pairs that illustrate the general idea for the design, and some key ideas. I'm hoping to get them done by the end of December 2016.

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 :)
------------------------------
------------------------------

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...

Ruminations on Everything and Nothing

Common relevance - situational assumptions.

The common-use ontological notions of nothing/emptiness are context dependent, with enthymematic information filling out the gaps concerning what kind of absence is under consideration, and as such relevant absence. This isn't very different from notions of universal nature, such as all/everything, where what counts as anything at all, is conditioned on the relevant context. I will argue that the empty class is a relative notion, and is conditioned by what is considered as possible/impossible in any given context, whether it be an informal one (situation) or formal (formal theory).

For example "she ate everything" (in most everyday cases) certainly doesn't refer to some paradoxical Leviathan who has devoured the entire world and would have to eat itself also, if the statement was to be considered as is, without qualification. Rather, almost always, the context dictates what the scope of quantification is restricted to - and in most cases the type of food stuff contents of, say a plate, or fridge etc.

Similarly in the cases of common expressions such as "the house is empty", or even ones like "the house is absolutely empty", or "there's nothing left" the scope of quantification is restricted to the things that count as ontologically relevant e.g. people, furniture, etc.

Relevant scope of quantification can be thought of as relevant possibility (context conditioned/indexed possibility). When I say "Put down your pencils everyone!", in a room full of students somewhere in one of the University of Queensland St. Lucia campus lecture rooms, at the end of a multivariate calculus quiz, this address does not include the security guard checking id's in the Biological Sciences Library, nor does it include my friend who at that very moment is sailing the Mediterranean Sea - they're outside the scope of relevant quantification, since they're not even enrolled in MATH 1052 (arguably, a non-controversial minimal requirement). This would still hold if by some strange circumstances my sailor friend somehow acquired a copy of the quiz and was solving the problems simultaneously with my students. Not being a relevant someone in that context, she is a relevantly impossible someone. 

Of course there is a world where she arranges with the school to be allowed to take the test remotely, and as such participate in the quiz via some appropriate video-phone technology. However in most cases of common language use, such peculiar circumstances do not tend to be part of enthymematic content (common situations don't imply uncommon information about them, i.e. our reasoning in most cases is governed by a ceteris paribus set of assumptions), but rather carefully explicated instead.

Furthermore, if after 15 minutes or so, having marked "all" (the ones handed out in the classroom) the quizzes, I said "No one got full marks.", and if it turned out that my sailor friend in fact got all the answers right, I would still be telling the truth, for my friend fails to be in the scope of the relevant quantification (indexed by that informal context). 

Formally, we can think of my friend as being an element of the empty set (here indexed by the above quiz scenario), as far as "someones" are concerned. But since empty sets have no elements, instead we think of the sailor friend being in some sense associated with the empty set indexed by the quiz scenario. Since such relevantly impossible objects can be thought of as a collection, we can also talk of the cardinality of that collection. But since we're interested in a particular relation such a collection has to the empty set indexed by the quiz context, rather than using the term cardinality, we use the term rarity, which captures the intuitive meaning of the cardinality in question.

Meta relevance - logical assumptions.

There seems to be no reason why the above arguments, and their form, cannot be generalized, and as such extended to formal languages, logical systems and their theories. That is, what counts now as relevantly possible in some theory (a theory being the situation analogue of the above quiz scenario), is just what is possible in the logic on which the theory is based.

A simple idea.

Physicists disagree about what exists, but they tend to agree that nothing is just the lack of that which exists. Note that this is physical nothing, or physical impossibility. Nothing "contains", as it were, all the physically impossible objects, but there are none, so nothing is "empty", as it were. There is one more interesting point to add, which will play a crucial role later on - sure physicists will agree that nothing is the lack of any physical thing, but since what counts as a physically possible object needn't be congruent across physical theories (it's not), physicists will disagree about the list of stuff that nothing is meant to denote the lack of.

Is it really that difficult to extend this idea to logical impossibility? I think not. Analogously, we extend the idea of physical nothing to logical nothing as the lack of what can exist. Here, nothing "contains", as it were, all the logically impossible objects, but there can't be such things, so nothing is "empty", as it were. Furthermore logicians will tend to agree that nothing possible is the lack of any possible thing, but since what counts as a logically possible object needn't be congruent across logical theories (it isn't), logicians will disagree about the list of stuff that nothing possible is meant to denote the lack of.

Clearly, in some set theoretic ontological model nothing possible can be identified with the empty set. That is we think of the empty set Ø_K(L) as the object associated with all the impossible objects in the L theory, K. We can thus speak of the order of the empty set, here called the rarity, defined as the cardinality of all the K(L) impossible objects.

Illustrating some of Ludwig Wittgenstein's ideas from the Tractatus.

Zeno's Apology

We know that the ancient Greeks loved their wine. So must have Zeno, the Greek philosopher who among other things is known for devising numerous paradoxes intended to endorse the views of Parmenides.

One of those paradoxes, called "The Dichotomy" claimed that you could never traverse a finite stretch of space, for in order to get to the destination one would have to first get to the half-way point, but before that one would have to reach the quarter-way point, and so on. So since traversing a finite stretch of space entailed performing an infinite number of tasks, reaching the destination, Zeno argued was impossible. 

We could rephrase this paradox in a way to claim that finishing a goblet of wine is impossible. For before one drinks the whole lot, one need's to drink half first etc. 

Now, either Zeno was also a very unusual Greek insofar as he abstained from wine - which is very doubtful, and would surely be insulting to even assume - or when it came to wine drinking he could perform the impossible! 

I'd take that as a valid defense, if Zeno chose to use it.

Much Ado About Nothing

AN EXERCISE IN INTUITION
Suppose we have two physical vessels of varying size, U and u, and suppose further that they're both empty (in the usual sense - we've run out of wine). Should we consider the degree of their emptiness as the same (i.e. even talk of such degrees is nonsense!), or can we say in a meaningful way that their degrees of emptiness differ?

AN ARGUMENT
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.

The above argument, and exercise in intuition do not exactly express the same ideas, but overlap on what is essentially the general idea.

Mad Hatter Paradox.

DEFINITION: Mad as a Hatter  – someone is said to be mad as a hatter iff there exists a mental illness (madness) from which they suffer, and they’re ignorant of its existence.

Now, consider someone asserting “I know that I’m mad as a hatter”. Call it the Mad Hatter sentence, and denote it with MH.

Is MH true or false?

Suppose MH is true. So it’s true that the person uttering it knows that they’re mad as a hatter. But by knowing that they are mad as a hatter they’re aware of the illness they purport to be suffering from, i.e. being mad as a hatter and so by definition fail to satisfy the conditions for being mad as a hatter. Hence MH is false, it seems.

But if they are not mad as a hatter, and assert a knowledge of being such, the person asserting MH is oblivious of that ignorance (them in fact not being mad as a hatter), a delusion of sorts, and hence qualifies them for being mad as a hatter, thus rendering MH true - but we know where that leads.

My independent discovery of the sequence A007526

I came across a rather surprising property of the above recurrence relation which I defined when counting the number of models of conditional logics with Lewis-Stanlaker type semantics, i.e. similarity spheres. Because the structure of nested sphere assignments happens to correspond in its form to permutations of non-empty subsets of some set, the formula, known to combinatorialists since Bernoulli, had emerged.

Below is an example taken from The Encyclopedia of Integer Sequences: sequence A007526 entry.

I say those properties were surprising given the context, for in and of themselves they’re not surprising at all. I simply didn’t expect such interesting properties to be inherent in the structure of the systems I was studying.

The sequence s is related to the number of models, given certain parameters (more precisely, the parameter is the size of the filter which is the set of all subformulae of a formula whose satisfiability is being tested - hence the requirement to generate all and only those models which are sufficient and necessary for that task), and given by the following recursive definition:

And this is the "curious" result, which links the above recursively defined function with Euler's Constant e:

It's easy to show that the explicit solution to the above recursive definition of s is given by the formula:

And considering the well known Taylor series, it's easy to show that the result stated above follows.

It turns out that this is a known sequence A007526., whose earliest references are Izquierdo (1659), Caramuel de Lobkowitz (1670), Prestet (1675) and Bernoulli (1713). So it appears that I've discovered/defined this sequence independently of those great mathematicians.