The Abyss and the Tao

After Cantor recovered from a stint in the insane asylum for mental breakdowns suffered while trying to prove some (we now know) intractable problems in set theory, he had a stroll with his good friend and colleague Dedekind. Dedekind asked Cantor what he pictured in his mind when he thought of sets. Dedekind remarked that when he thought about a set, he pictured a clear bag containing objects inside which are the set's elements. Cantor responded that when he thought about a set, he pictured an abyss.

In a previous post I remarked that some abstract objects occupy space and have relevant properties of concrete material objects and thus there may not be any philosophical (epistemological or otherwise) problems in dealing with them such as knowing them or having them cause and be caused by physical phenomenon. In mathematics, we have pure sets built from the null set from iterative or otherwise “mental operations.” The old joke that the mathematical universe is an entire infinite universe begotten from nothing (the contents of the null set) is illustrative of the counter-intuitiveness of this idea.

Thus, as the story goes, from the null set, we have everything we need for mathematics. But since these objects of math are all purely abstract without concrete properties (as opposed to abstract objects with concrete elements) which may take part in the causal epistemological chain, how do we know them if e.g., the causal theory of knowledge is correct? If mathematics is built on such foundations, we cannot appeal to impure sets such as sets built on singletons of everyday objects to resolve these epistemological lacunae. Mathematics foundationally built in such a way thus may be metaphysically and epistemologically counter-intuitive to many precisely because of the knowledge problem or some other problems with causally inert properties of pure sets or the begetting of things from nothing. Metaphysically, it is problematic as the old joke suggests, it is a infinitely high house of cards built on top the foundations of an abyss. I want to ask here if there is another alternative system that is built on firmer and more intuitive metaphysical foundations.

Now consider the classical Chinese conception of numbers. Many philosophers of the school of names and the neo-Taoist school inspired by the I-Ching (such as Wang Bi) thought that from some object, we may, by some mental operation, form the class (they did not have a modern notion of set obviously but did have a notion of what we would term 'class') containing that object. Now we have two objects, namely the object and its singleton class. One can go on forming more and more objects this way until one has all the objects needed for one's numbers and thus to furnish one's mathematical universe. This would generate the positive integers instead of the natural numbers including zero as modern mathematics would have it. There is something to be said about the Chinese system; it is iterative like the modern conception of numbers but it starts off from one (the singleton set of some concrete object) instead of zero (the null set). This system is ontologically well-founded on something as opposed to nothing and is concrete "from the start." Specifically, what the Taoist founds his theory of numbers on, the original object on which all his classes are built on, is just the Tao.

More formally, we have according to the Taoist conception: Tao, {Tao}, {Tao, {Tao}}, {Tao, {Tao}, {Tao, {Tao}}}. Now, either the Tao or {Tao} may be arbitrarily designated as the number one.

Let's say that we choose the former. One is unlike the other numbers in that 1. it is not a class (set). 2. for all numbers n other than one contains n elements while one, not being a set contains no elements.

Or one can designate the Tao's singleton {Tao} as the number one. But then all subsequent numbers will contain the Tao which is not a number to generate all successive numbers.

Alternatively, one may have 1=Tao, 2={Tao}, 3={Tao, {Tao}}.

This last interpretation seems to be the one Wang Bi favored.

Significantly, as Wang Bi makes the point in both his Yijing and Laozi commentaries, in this sense “one” is not a number but that which makes possible all numbers and functions. In the latter (commentary to Laozi 39), Wang defines “one” as “the beginning of numbers and the ultimate of things.” In the former (commentary to Appended Remarks, Part I), he writes, “In the amplification of the numbers of heaven and earth [in Yijing divination] … ‘one’ is not used. Because it is not used, use [of the others] is made possible; because it is not a number, numbers are made complete. This indeed is the great ultimate of change.”

However, here, each number will contain n-1 elements instead of the Von Neumann formulation of modern mathematics which has for each number n, n elements.

Thus Chinese mathematics can be impure and would have no problems with the epistemological problem of knowing abstract objects built on the singleton of some concrete object. Everything that can be proven in the modern system of mathematics can be proven in the Chinese system per Lowenheim-Skolem theorem or one of its corollary theorems (since the two number systems are equinumorous and isomorphic). Zero thus is redundant for mathematics. There remains only orthographic problems of how to write numbers and mathematical formula down and the technique the classical Chinese used was to use a non referential “placeholder” in place of '0' or the cipher.

But this system is just as counter-intuitive as the modern for there are problems of its own. If I start off from the singleton set containing my computer and you start off from the singleton set containing your right foot, we would have two different notions of the numbers one and thus all subsequent numbers built on it. But 1=1 is true and it is necessarily true. Modern mathematics does not have this problem because it is easy to prove that there is one and only one null set and thus it and all numbers built on it are identical.

We may be able to obstruct this problem by introducing a single “arbitrary object” which can stand in for any object and starting from the singleton of it and define it as the number one. However, is this arbitrary object concrete or abstract? It might make sense to say of an arbitrary object that it is concrete or abstract with causal properties. I don't know. But if concrete where is this object? It may not have a specific location but since it could “stand in” for any concrete object, it may make some sense to ascribe to it causal properties even if it may be abstract much as the singletons of concrete objects. Kit Fine has defended arbitrary objects has having such common properties of concrete objects.

Alternatively, as pointed out above, Wang Bi and some of the other classical Chinese philosophers thought that the original object which one is is just the Tao, which in turn is itself not strictly definable and may be a ineffable "primitive" (I guess maybe like the term 'nothing' or perhaps 'arbitrary object') or its singleton. The Tao certainly seems to have causal properties and may certainly be said to influence causally the world on their conception whatever it may be.

Still, the Chinese system may not wholly escape some of the counter-intuitiveness associated modern math. It is only slightly less counter-intuitive than the modern system because instead of begetting the whole mathematical universe from nothing, it begets it from one thing. An infinitely high house of cards is built on the flimsiest of foundations instead of on an abyss.

Hate speech

Hate speech is Constitutionally protected. Many proponents of a liberal democracy see no wrong in banning hate speech because they think that hate speech violates the rights of others and thus violates some sort of harm principle. In a liberal democracy, the idea is that one can say, think and do anything one wants so long as it doesn't harm others (especially violates some of their rights) but it has been argued that hate speech violates this principle of liberalism. Some have argued that the harm principle itself is anti-liberal at its core and so one may not appeal to it within a liberal framework to justify the banning of certain kinds of speech. However, there seems to me to be another reason or reasons to ban hate speech within a liberal framework.

Think of the most common restrictions on speech in a liberal democracy: commonly cited examples include yelling "fire!" in a crowded theater and defamatory speech. One might be able to assign rights to whole groups of people (as international and domestic law sometimes do). One of these laws may be analogous to anti-defamation laws protecting individuals against libel or slander. One way to look at hate speech is that it is defamation against whole groups. Thus if we can justify anti-defamation laws in a liberal society, we can justify anti-hate speech laws if hate speech is analogous to defamation. The most obvious retort is that they are not analogous.

Some hate speech clearly are similar to defamation such e.g., spreading the lie that Jews are involved in a global conspiracy to kill all "Aryans." That is a claim, a false assertive claim about a group of people made to defame that group and comes at considerable costs to that group and thus may be liable to analogous legal repercussions. But consider a racial slur, "nigger." Referring to someone black using this term is not, on the surface, making an assertive claim about her or her racial group and thus a fortiori, not making a false defamatory claim. So on the traditional interpretation of racial slurs, "That nigger, John was fired" would mean the same as "John was fired" or "That black man named, 'John' was fired."

But is that really the case? Some philosophers of language such as David Kaplan and Christopher Hom claim that certain nouns including racial slurs do have assertoric content. Hom's paper "The Semantics of Racial Epithets" is a classic work on this topic. For racial slurs such as "nigger," Hom argues that it is translatable to a conjunction of sentences (or propositions or whatever it is that have semantic content) that makes pejorative claims about a group or a person belonging to the group. The longer and more institutions of hate associated with that slur, the more semantically "explosive" or harmful it is because it make more implicit assertions that are packed into the slur. When used, the slur becomes "unpacked" and explodes all the (defamatory) assertions associated with that group within some society in which there is a history of institutionally associating negative stereotypes with some group to the slur.

Infinite lives

Imagine a race of immortals. Nothing can kill them; not disease, old age, etc. They are invincible and they also do not go through the normal aging process of decrepitude of the body and mind associated with mortal aging. If they live forever and these beings are like us in all other ways, they will go through the normal vicissitudes of life, the ups and downs, highs and lows, good times and bad times. So such a life in total will include an infinite positive utility and an infinite negative utility. No matter how good a life one lives, so long as some significant repetition of periods of bad times, they will go through an eternity of suffering as well as good times. Do these times cancel out since both times in bad and in good are infinite in duration? Will such a life ever be worth living if we are to believe that what makes life worth living is experiencing as much good (defined as pleasure or happiness here or "utility") and as few bad (sadness, misery, suffering, etc) as possible? In a finite life, once can experience far fewer bad times than good and vice versa but an infinite life, all experiences, good and bad are equal in amount: both infinite in duration.

Also consider that such an immortal, no matter how virtuous he or she is unless she is perfectly virtuous, will inevitably commit an infinite amount of evil as well as good. Will such evil acts and thoughts be canceled out by her good acts and thoughts? It would appear so as well.

Now consider the human race. If the human race lives on forever (and there is almost no chance of that) then the human race will achieve the same amount of overall collective good and bad. It will achieve (unless it all becomes perfectly virtuous after some period of time continuously) an infinite amount of evil and good collective virtue.

Movie suggestion

If you haven't gotten a chance to see Moon, I highly recommend it. Those who are familiar with the literature will see all the references to personal identity. The director was once a grad student in philosophy.

Meaning of life again

I posted earlier that what I take to mean when people talk about the "meaning of life" is either the value (or highest significance, etc) of life or, on the other hand, the purpose of life. But sometimes people may have another meaning in mind.

Some may think when they say things like "life does not have meaning" is that there is no non arbitrary or objective narrative structure to life, no grand story of life. More specifically, I've heard that many people who believe in god may think that life has a "narrative" structure created by God and that it is our job to find out what this narrative structure is. It is there independent of our interpretation. It is a grand story of life, complete with beginning, middle filled with some kind of conflict (between good and evil etc) and teleological end. I will call this the hermeneutical understanding of the meaning of life. Finding out this meaning is analogous to finding out the correct interpretation for a Shakespearean play only we are the actors and stage directors that are to be in it. Shakespeare may very well have intended his plays to be interpreted in certain ways as opposed to others. Knowing it helps us to know our roles well perhaps: how to act, what to say, how to say it, etc.

Some people would be deeply upset by the fact that there is no such absolute hermeneutic meaning if it turns out revealed that there's no god-author and they will not be consoled by the existentialists' advice that we should, in some sense, be the authors of our own life story. Those not satisfied with that advice may see this as arbitrary or a "projection" without the absoluteness if divine meaning or some kind of absolute principle that we don't impart on reality.

This kind of thinking makes a linguistic analogy (hermeneutical). However, this understanding of life's meaning simply pushed the problem one step back for even if there is a god and he has intended life be lived and interpreted in a certain way, we may still ask if that way of understanding and living is absolute or arbitrary. In other words, there is a "further question problem." Notice this is analogous to the problem Plato posed about morality in the Euthyphro. As Socrates allegedly said:

"Is the pious (τὸ ὅσιον) loved by the gods because it is pious, or is it pious because it is loved by the gods?"

This is a moral direction-of-fit problem. Philosophers seem to universally agree that if there is a god or gods, they cannot choose to make whatever they want moral (or pious) much as they cannot make it so that 2+2=5. Anyone, even if god, doubts the truth of that mathematical claim seem not to understand mathematics or be communicating in a different language from us when they doubt that claim. Analogously, they cannot make it so that torturing babies for fun is right. If torturing babies for fun is wrong, it is wrong whether the god(s) believes or loves the fact that it is it is or not. If anyone denies that moral fact, it seems they also not know what moral terms and claims mean or they seem to be using a different language altogether.

If god made the world with an intention in mind as to its hermeneutic meaning, we can always ask if this meaning is arbitrary and if god could have made it otherwise. If it is arbitrary then it is, in a real sense, not absolute and we are back to the arbitrariness of life. The theist does not escape arbitrariness in meaning by appealing to god. If god put in some meaning because it is valuable and not because it becomes valuable by the mere fact of him endorsing it, then god simply factors out of the equation. It is valuable on its own. Even if god had never lived, it would still be valuable to live life according to that interpretation.

Abstract objects and causation

In another post I argued that there seems to be some reason to think that some abstract objects occupy physical space. A (impure) set containing some material object(s) for example.

There are classes of problems in philosophy that this conclusion may be related. Consider the causal theory of knowledge (I believe first proposed by Alvin Goldman). One of the objections to this theory of knowledge is that it can't seem to explain how we come to have knowledge about abstract objects like sets and numbers. But if coincident objects (say, a set containing a car as only element and the car itself) can be thought as two objects which completely overlap spatio-temporally, then can't we think of these two objects as affecting other objects (physical or abstract) when they interact with them? In other words, if some abstract objects occupy space (compare the Cartesian concept of "res extens") like physical objects do, can't they also interact with physical objects? Doesn't it make sense to say that both the car caused the accident or, (albeit awkwardly) that the car's singleton caused the accident?

If so then at least some abstract objects can cause effects in the physical world. But if it makes sense to say that some sets may cause effects in the physical world, why not all? Other problems with dualism which asks how the mind, if it is not material, can cause physical effects by controlling the body, etc may also be less problematic once this once thought problematic issue is not thought so problematic. Relatedly, the ancient Nyaya philosophers of mind, as far as I know, had no problems with abstracta being extended in space or time such as their conceptions of the mind. That may be why they didn't seem to have much problems with mind-body causation despite being dualists.

Wagering for one's soul and the possibility of Divine Evil

In the previous post, I mentioned that the “God” of the Bible is at least a possible, if not actual being that is very evil. If the description of him is empty, he is a possible being of some possible world (closest to ours with relevant changes) that fits the description. If he is an actual being and “God” of the Bible denotes him, the description may or may not be accurate. The Bible may be wrong about his attributes. He may not be the evil being he is described. In which case it may be that people have mistaken his attributes or it may be that he is actually an omnibenevolent being that has constructed the Bible as a kind of moral test for us. If we resist the Biblical version of him which is on many philosophical accounts is a description of an evil being, we “pass” and if we fall for it by worshiping the Biblical description, we “fail.” What is the consequence of these possibilities?

Consider these four possibilities. Either there is at least one evil being that fits the Biblical bill (henceforth, I will refer to this possible being as "Dog") or there is not. Additionally, either there is at least one omnibenevolent and omnipotent being or there is not (I will call this being just "God" not to be confused, of course, with the Dog which is the Biblical God). There are four total possibilities and they are mutually exclusive: there are at least one Dog that fits the Biblical bill but no God. That both exists. That God exists but Dog does not and that neither exists. In what follows, I will simply ignore the possibility of more than one of each kind of being (it will not affect the argument).

Many philosophers since Augustine have made maltheistic arguments against the Biblical (Abrahamic) God. If we are to interpret the Bible literally, we find that it's inescapable the conclusion, if we are rational, that the being called 'God' there is actually something monstrously evil. His actions in the Bible of causing/inciting/endorsing/ordering genocides, mass murder, "natural" catastrophes, child and wife killings, etc are well known. Even some of our Founding Fathers have remarked how despicable and evil the Abrahamic god is. Jefferson once said (paraphrasing) that a unbiased and reasonable man would have to conclude that the Biblical god is evil or a demon after reading it.

David Lewis in his paper, “Divine Evil” has gone even further in suggesting that the evils perpetrated in these well known examples from the Old and New Testament pale in comparison to the infinite evil of Divine Evil (for all former evils are finite while Divine Evil is infinite). Divine evil is the evil of punishing those to an eternity of suffering (damnation) for facile reasons such as vanity. Yet it is precisely for reasons such as vanity that the Biblical God has given in the Bible (and Koran) for sending people (many of whom presumably lived very virtuous lives) to an eternity of suffering. Making someone suffer infinite torment for an infinite amount of time for facile reasons constitute evil par excellence. It is hard to even imagine what could be more evil. It is infinite Divine Evil. I wholeheartedly agree with this Divine Evil hypothesis, that is, that the being described as the God of the Bible appears to be very evil.

Lewis uses the hypothetical example of a Nazi named "Fritz" to argue his case and who does not act in an evil manner but wholeheartedly supports the Nazi cause of invasion, aggressive wars, and genocide. He admires Hitler very much. Now most of us I think will consider Fritz very evil even if he has not done acts of evil. He is evil simply in virtue of endorsing and advocating those evil acts. He is vicariously evil. That was Lewis's main argument, that people can be vicariously evil in regards to obeying the Biblical commends to worship the evil Biblical being that sends people to torment for eternity for vain reasons much as Fritz can be evil just by admiring and endorsing Hitler. Notice that this vicarious evil need no real object to imbue it with its evil; even if Fritz is a brain-in-the-vat and the Nazi leaders and desired world he endorses is an illusion created by some neuroscientists, he would still seem to be evil in endorsing such evil illusions. Fritz, after all, has no control over whether his world is real or an illusion and if he is guilty of evil in a real world, he is so in a illusory world which is phenomenally identical to that real world. Whether he lives in a real world or an imaginary but phenomenologically identical world is a matter of luck outside of his control and which he presumable is not responsible for.

Assuming that Lewis's argument for the Divine Evil of the Abrahamic God is true, what has this got to do with us? Other than what it has to our moral status there seems to be prudential concerns we must also deal with analogous to Pascal's prudential arguments for believing (or trying to cultivate a belief) in his god.

If the the Abrahamic God is such an infinitely evil being, what prudential arguments analogous to Pascal's wager can we devise to take into account such a conclusion?

In the following table, I will argue in similar decision theoretic terms for what we should believe or disbelieve and maybe be against (that is, cultivate a antipathy or something like it) so that we may avoid eternal damnation and end up with the best chances at getting into heaven.

I have also avoided tricky technical issues with interpretation of the wager and with issues between infinite and finitary decision theory. Alan Hajek employs interesting methods using kinds of ordinal infinite numbers called surreals to remedy the usual problems associated with infinitary decision theory such as problems with mixed strategies (i.e., since run of the mill infinity is reflexive under multiplication of finite probabilities, any mixed strategy will yield infinite utility thereby making a mockery of Pascal's wager), and infinitesimal degrees of beliefs and Roy Sorensen uses more traditional conceptions of (cardinal) infinity to make the wager as fitting as possible to the tasks at hand.

One may also employ very large finite utilities to stand for heaven (and inversely, large negative utilities for hell) but I prefer the infinite strategies as it is closer to our religious and philosophical notion of the utility of heaven and hell but other problems with the nature of non standard uses of infinite numbers inevitably rear their heads making trouble for Pascal's interpretation of the wager as Hajek shows. Since juggling infinite utilities seems to cause technical trouble (tractable or intractable is a matter of dispute) so I will try to avoid such technical issues with the ambiguous utilities of -H (which is hell and may or may not be infinite negative utility) and +H will mean heaven (which may or may not be infinitely positive utility). I think the wager can still be analyzed rationally using just the notion of dominance from decision theory.

-H and +H, even if finite, will likely be under any plausible interpretation, very, very large utilities in opposite directions. One is to be avoided at nearly all costs and the other, desired at equally far reaching costs. I will use “Indeterminate” to be somewhere between these utilities as it is indeterminate what our reward/punishment will be under some such scenario. However, indeterminate leaning towards some direction (such as -H or +H) as it may be will lean towards that direction. So we have the current utility spectrum from least desirable to most (for us).

(-H) is less than (Indeterminate) is less than (+H)

I will use the palindrome "Dog" to denote the Biblical god and assuming that Lewis's argument for its evil is true (and in fact, it may be a sort of analytical truth that a being displaying those qualities are evil as I think Lewis would agree). I will also use "God" to denote a possibly existing being who really is omnibenevolent, omnipotent, omniscient, etc; this is the being with the properties traditionally thought to belong to God's. Thus my terms are:

Dog=Abrahamic/Biblical god=Devil

God=Omnibenevolent, creator...etc

I have assumed (much as Pascal and Hajek's versions of the wager) that your subjective probability for all these possibilities are positive (or all >0). The first three columns of this table denotes the possible situation when God is all powerful with ability to send people to heaven/hell but Dog does not have such powers, and this would correspond with many of our understanding of God and the devil. Dog may claim to hold such powers to be able to send people to heaven/hell as he does in the Bible (if the Divine Evil conclusion is true as we are assuming). The fourth and fifth columns represents possible scenarios where both God and Dog have powers to send us to -H or +H. We have the following 2 by 6 decision matrix:

	God & Dog exists	Dog only	God only	God & Dog exists (both equally powerful)	Dog only (Dog has power to send people to -H)	Neither exists
Wager for Dog	-H	Indeterminate	-H	Indeterminate (but leans towards -H)	Indeterminate (lean towards -H)	Indeterminate
Against Dog	+H	Indeterminate	+H	indeterminate	-H	Indeterminate

The columns represent these exhaustive possibilities. Either both God and Dog exists, or Dog exists but God does not or God does and Dog does not or neither exists (last column).

When God and Dog exists but Dog lacks powers to send us to -H (first column), it is prudent to wager against Dog. When Dog exists without God but Dog lacks the requisite powers to send us to -H (merely bluffing to, e.g.) as is the case in the second column, the results are indeterminate much as what happens when we die in an atheist's world. When both God and Dog exists and both are equally powerful, it is indeterminate who will get our souls but I am leaning towards -H for those who wager for Dog as both God may want to punish us for our vicarious evil in wagering for Dog and Dog may want to punish us just because he is Dog. He may repay our good turn for our wagering for him with an evil turn. He is evil after all! In that case, both God and Dog will agree on our fate. But he may also wish to reward us for our wagering for him but God may have other ideas as he may wish to punish us. Anyway, I am leaning towards Against Dog as the rational choice in this scenario even though I have labeled it indeterminate what our fate is if we so choose for that reason.

In the fifth column, only Dog exists and he has the power to send us to -H. If we wager for him, he may show us some good will and not send us to -H. But he may also lie (again, he is evil after all!) and decide to send us there anyway against his promises in the Bible to send all faithful to +H. If we wager against him he will certainly send us there.

The last column is where neither God nor Dog exists and in which case we just die and what happens to us is indeterminate or likely we just get a utility of nothing.

Notice the asymmetry. If we ignore the fifth column where Dog has the power to send people to -H and there is no God to check his powers, we have a decision matrix that is dominant for “Against Dog”. That is, if we ignore the possibility that Dog is the only power in the heavens without God to check his evil ways, we would be irrational not to wager against Dog. Even if that columns is true, we may only have little prudential reason to choose to wager for Dog as he might deceive us and send us to -H anyway. It makes sense that the devil pays a good turn with a bad one so given this there is little reason we should decide to wager for him even if this column is in fact the case. This (especially combined with the moral reasons) suggests we should wager against Dog.