Science is Built of Facts

Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house.

Between the elements of a continuum there is a sort of intimate bond which makes a whole of them, in which the point is not prior to the line, but the line to the point.

Poincaré said so; and lines are not made of points in the way that counting numbers bigger than one are made of ones, although the number of all the points in a line will be made of ones in a similar way.

Nothing much...

Not much work done this week, too many distractions, such as a discussion About Faith (into which I managed to introduce some maths, though (incidentally, some of the deeper issues raised by that discussion are currently being examined at Truth and Purpose):)

My God!

My conception of God got (rather appropriately, if presumably accidentally) a perfect score on a test of its plausibility, due to my ticking just 3 boxes: (i) my God is the "Creator (of all that exists)," (ii) a "Perfectly Free" agent, and (iii) "a being with whom one can have a personal relationship."
......Do-It-Yourself Deity heads this list of games at TPM, and said of my conception:

The metaphysical engineers are happy to report that, to the best of their knowledge, the God you conceive is internally consistent and could exist in our universe. But they are less sure that what you have described deserves the name of God. She is not, for example, all-powerful. A God which knows everything or is totally benign may be a wonderful ideal, but is she really a God unless she has ultimate power?

We suspect that your God is not the traditional God of the Christian, Jewish or Muslim faiths.

Not traditional? The creator of this Universe, and a personal God? Whatever...
......Anyway, had I said that God was "Omnipotent (all-powerful, able to do anything)," I'd have had the problem (according to the engineers) that S/he couldn't make 2 + 2 equal 5; and of course, nor could S/he have made it so that S/he was always nothing, but so what? Surely a perfectly free person with absolute physical and spiritual power over at least this Universe would be a God... Presumably, with such a reading of "all" I shouldn't have included "Creator," since I don't think that God must be self-created (which I suspect is impossible:)
......Incidentally, the reason why I excluded "all-loving" was the possibility of including evil acts in the quantification; but the engineers missed that, and said instead that I'd have to give up the personal relationship, in that case (I can have a personal relationship with someone only if there's something that S/he doesn't love?)
......(And why were the implausibilities of omniscience overlooked, e.g.:)
......Incidentally, I'm not religious, as such (once a Methodist, I discovered that I hate singing), but if I was going to be then according to this test I should consider:
1. Reform Judaism (100%)
2. Liberal Quakers (96%)
3. Bahai (91%)
4. Unitarian Universalism (86%)
5. Mainline - Liberal Christian Protestants (80%)
6. Orthodox Judaism (78%)
7. Sikhism (78%)
8. Neo-Pagan (75%)
9. Mahayana Buddhism (73%)
10. Islam (69%)

Is there a Point?

I've actually managed (accidentally) to do something relevant this week, as I'm currently commenting on an argument that points don't exist...

Why Phi

In a shift of historic importance, America's colleges and universities have largely abandoned the idea that life's most important question is an appropriate subject for the classroom. In doing so, they have betrayed their students by depriving them of the chance to explore it in an organized way, before they are caught up in their careers and preoccupied with the urgent business of living itself. This abandonment has also helped create a society in which deeper questions of values are left in the hands of those motivated by religious conviction - a disturbing and dangerous development.

Anthony Kronman, in Why are we here? (16/9/2007, The Boston Globe). Theology is as organized as philosophy, though. And what do philosophers actually do, anyway?

Science and Omniscience

When Patrick notices that he has overlooked something, he knows what it is like for him to notice that, but it seems odd that an omniscient being should know that... Unless that being was Patrick's creator (to whom he need be no less transparent than his own conceptions are to him) perhaps; but did Patrick overlook anything?
......Let SDL = God doesn't believe that SDL expresses a true proposition. On page 145 of The being that knew too much, Patrick argued that a rational person who believes that SDL is nonsense, and also that God believes only truths, would have to believe that God doesn't believe that SDL expresses a true proposition, whence (since that appears to be SDL) that rational person would believe that there was a truth (expressed by SDL, apparently) that God doesn't know... but appearances can be deceptive. Why should any rational person believe that nonsense could express a true proposition? (I argue here that sentences like SDL are nonsense, and as Michael said:) “You can’t say you are talking nonsense by talking nonsense, since to talk nonsense is not to say anything. But having talked nonsense, you can go on to say that it was nonsense, and now you are talking sense.”
......Anyway, bad arguments against omniscience (e.g. those that work only within set theory) aside, the following is an argument against it that goes (paradoxically) via a possible motive for the creation of a world not unlike that described by our best-tested science (an argument given more briefly earlier): Something that no sapient being (or telepathic beings, etc.) could possibly know is the full extent of what s/he doesn't (they don't) know. How could s/he rule out, for example, the existence of similar beings of which s/he was unaware? Maybe there are none (and if some, then there is something s/he doesn't know), and maybe s/he even believes that there are definitely none (although that would be foolish), but how could s/he know that? No one can have empirical access to places that are completely cut off from those s/he inhabits (by definition), so s/he could not know empirically that such places (which s/he knows are of a possible kind, since s/he inhabits one) are not inhabited by such beings (of which kind s/he is one), and how else could such propositions (about what would be, by definition, an external world) be known?
......So, even a relatively perfect being (which is, after all, the sort that we might expect to exist, much as we might expect a unified theory of physics to be elegant) might know that s/he could not be so absolutely omniscient. And rather than remaining ignorant, it would surely be better to be able to create ways of probing such unknowns (for which s/he would need to be able to change, and so exist in something like time, but since this would be better, that need not be a privation), whence s/he might want to explore such possibilities (as much as possible) by creating opportunities (insofar as s/he is capable) for such possible others to act (in some place to which s/he would therefore have empirical access:)
......Such a creator would therefore be perfectly compatible with all we know scientifically (and also religiously (to a less definite degree (naturally)))... A list of agreements would therefore be tedious, in a blog (here I'd rather ask, what is incompatible with that hypothesis?); but in short, if it was me I'd want first to find the limits of my natural place, insofar as I could discover such limits (and if not, I'd want to invent some), so that then I could make something like a bridge there (if I didn't discover something like that there already), something like a blank sheet of paper—something, in short, like this Universe, in which relatively simple physical laws allow a wide variety of agents, with little need for intervention (whence atheists note that if there is a God, he must be lazy). Of course, it may well be that there are no such others, that the quest would be endless; but therefore I'd make my paper (so to speak) beautiful, and recyclable (e.g. by having my creatures all return to my place for debriefing:) One day my walls might be found scribbled upon though, so I'd've built them to facilitate a wide range of responses too...

Arguments for atheism

Religions are crazy.

So are scientific worldviews: are you a robot,
a biochemical robot whose future already exists
in the future part of spacetime?

I cannot see God.

Can you see ten dimensions in the universe?
I only see three in the space around us.

Clever people are atheists.

Clever people are bankers and lawyers.

And so forth.

For more, see the carnival of the godless.

A Bad Argument (for atheism)

A famous sceptical scenario, BIV, asks you to imagine a disembodied brain (otherwise just like yours) in an alien scientist’s laboratory, being stimulated in such a way that that person (who’s brain it is), say X, seems to be in a world just like the one around you... and then asks, how do you know that you are not that person, X? The scenario is regarded as a sceptical one because it seems (the brain being disembodied) that X would not have real hands. But I think that X would have real hands if X was you; what would not be so real (as the things around us, by reference to which we all correctly interpret the word “real,” within our language) would be your brain (the one in the alien’s laboratory)—and so your brain might rightly, were you X, be thought of more or less as you now think of your soul (about which we know little). To ask, are you in such a BIV world? is to ask, do you have a soul? And to ask, does this Universe have a Creator? is to ask, is there such an alien scientist? But not even Dawkins would argue that since our hands do exist, therefore God doesn't!

Continuity, Unity and Infinity

I’m back in Glasgow wondering, what are continua? Things that extend smoothly, is the obvious answer, but what is extension? And while things presumably extend smoothly if they contain no gaps that contain something else (whence they are unities), what if the space through which they extend is discrete? So what makes intuited space a continuum (so to speak)? Whilst we think of it as extended because it contains many things like our own body (e.g. many users of our own language), so might something more discrete (e.g. a finite matrix). So presumably another criterion is that those parts of a continuum that are not points must all be endlessly subdivisible (whence continua have infinitely many parts). But that is insufficient, as the rational number line is not thereby excluded, from being the shape of space itself (even though it does not contain the diagonal of the unit square)... (To be continued:)

Science is built of facts

and so is skepticism ...

It is cloudy. Trillions of droplets float in the shade of other droplets,
getting heavier and heavier until eventually they rain down. The sun
breaks through the shrinking clouds, and the sky frowns in a rainbow

(sunlight enters the drops, refracting as it does so, and when it reaches
the other sides of the drops it is reflected back; when it leaves the drops
it refracts again, and heads down to my eyes: I see a rainbow up there).

Under that grey sky, the green of the sunlit trees stands out, seeming
to be where the green trees are. But I know that that greenness is in
my mind, where the green of the rainbow is. Where are the trees?

There they are; but I put that greenness out there, so I wonder
did I put the trees there too? I take this world seriously but
I take my nightmares seriously when I am having one.

What Katy Did

Once upon a time, Andy found a caterpillar, called it Katy, and showed it to his friend Bobby. Eventually Katy turned into a butterfly. “That’s Betty,” said Bobby. “No,” said Andy, “that’s Katy.”
......Now, is there a fact of that matter, about whether or not a caterpillar, and the butterfly that it changes into, are the same individual? I don’t see how there could be. On the one hand they have the same genetics, but on the other hand they have very different brains. And consider how, whereas a zygote divides into two cells of the same individual, an amoeba divides into two different amoebas, although maybe it would not be too unreasonable (if a little unnatural, from our human perspective) to regard that division as the original amoeba getting itself a disconnected multicellular body. Et cetera...
......But we’re rushing ahead of ourselves; so let’s return to Andy showing Katy the caterpillar to his friend Bobby. If names refer directly to the named objects, there must have been some definite object there, to be called by that definite name, ‘Katy’. Both Andy and Bobby knew (we may suppose) what caterpillars are (and what butterflies are), so that object was presumably the caterpillar in question—there was, of course, only one caterpillar there, Katy (and only one language being spoken, English), but then, for example, either that individual, Katy, had the property of sometimes having wings, or it did not. In short, either Bobby was wrong about the butterfly, or Andy was (? Although it’s probably just my failure to grasp the concept of direct reference, my intuitions being descriptivistic:)

Philosophy is Boring

“A detective novel written by a good philosophy student would begin: "In this novel I shall show that the butler did it." The rest will be just filling in the details.” (Jonathan Wolff:)

Brown is the Brightest Colour

Most mathematicians and philosophers writing in English think that the counting numbers are not properties of ordinary things like pairs of shoes, even though the view that they are not originates with a Nazi sympathizer, Gottlob Frege (1848–1925).

Frege was trying to prove that 1 + 1 = 2. He did not think that he could simply define 2 to be 1 + 1, because he did not think that numbers could be composed of ones. He thought that either those ones would be the same ones, so that one and one would just be one (much as you and yourself are just you), or else they would be different, a first one and a second different one, in which case the signs for them, 1 and 1, should also be different (much as David and David, when they are the names of two different kings of Scotland, are written David I and David II).

Could the signs for those ones be given subscripts, such as i and ii? But then we would have had to have defined ii already. As a nineteenth-century German professor of mathematics with very conservative views, Frege thought that he was well placed to devise new definitions of the counting numbers (and of the infinite numbers that had recently been discovered, by another German professor of mathematics, Georg Cantor), so he did just that. And then he attempted to prove that 1 + 1 = 2 with those definitions, impressing the far more influential Bertrand Russell, who went on to make his own infamously long mathematical proof that 1 + 1 = 2 with his own definitions.

As a matter of fact, 1 + 1 = 2 because one thing and another thing make two things. Those two things are different, but each of them is one thing, and much as the red of a rose and the red of an apple might be the same red colour, those ones are the same.

Much as red is a property of physical things like roses, apples, shoes and so forth, the number one is a very simple property of things: each and every thing is one thing. And numbers bigger than one certainly seem to be properties of collections of things (a pair of shoes is two shoes). And of course, the properties of physical things are described, not defined, so Frege had to find some good reasons why numbers bigger than one are not properties of physical things.

Frege observed that how many things a collection contains depends upon what kind of thing we are talking about. Two shoes are also a lot of molecules, for example. According to Frege, colours are not like that. A red shoe is not also some other colour, for example. Except in a different light, maybe. Or to a different pair of eyes. Or against a different background. And so forth; but in any case, he also observed that while we can say "the men are white" because each is white, we cannot say "the men are many" because each man is many (although each man is a lot of molecules).

We can say it because the men are many, though. Frege explained his argument in more detail (the following is a cheap and cheerful version of Frege's argument):

If you have a blue tooth, you could mentally remove the property of being blue to leave a colourless tooth. You could then remove the property of being a tooth, in order to see in your mind's eye what was left. You can, in short, mentally remove various properties, in order to isolate the other characteristics. But if you have, say, three teeth, and you remove all of the sensory properties of the teeth (their colours, their shapes, their hardness and so forth), would you be left with a one for each apple? Would you be left with the number three?

According to a very important German, Gottfried Leibniz, you cannot have three identical things. So, because you cannot strip away all of the sensory properties and be left with the number three, the number three is not a property of collections of physical things. What is mentally removing colours, though? Is it like looking at a black-and-white photograph? That makes sense, but would mentally removing their being teeth from some teeth be like having a hologram of some teeth? Or is it like having a description of some teeth, in some language? Or a mathematical model of some teeth? Or the logical possibility of some teeth? Or the idea of some teeth? Or the memory of some teeth? I find myself mentally picturing some teeth and then imagining that I am imagining them! Still, the argument above is a simplified version of Frege's argument. Let us stick with simple properties like colours and shapes.

Suppose that I remove the shapes and sizes of the three teeth, and their positions in space, but nothing else. Presumably there would still be something real. Would it be three teeth without shapes, sizes, or positions in space? But without shapes, sizes or positions in space, how could what was left be three of anything?

Perhaps it could be three overlapping wavefunctions. But prima facie, those teeth being three in number has a lot to do with their shapes, sizes and positions in space, all of which are properties. So, that version of this argument seems to show that numbers are properties of physical things (although by mentally removing all of the sensory properties of the three teeth, you might be left with three teeth in a space containing no sensory organs, in which case there would still be those three teeth). Frege's argument may well have been better (although Idealism was popular in his day), but it is certainly a simple matter to imagine any number of identical objects:

Imagine a space containing nothing but three identical spheres at the corners of an equilateral triangle, or four at the corners of a square, and so on. It is easy enough to imagine that all of the properties of those spheres, including their external relations with each other, are identical.

Perhaps there are better arguments that numbers are not properties nowadays (there must be some reason why the view that numbers are not properties is so popular with our experts, who do not usually agree about much beyond the bare facts, such as arithmetic). However, as I wade through the rather large literature that defends this rather unnatural view of that Nazi sympathizer, I am not expecting much. At first glance, a lot of the literature assumes that there are not really any collections of physical things in the world, just individual physical things. That is a bit like saying that there are not really any colours in the world, only photons, neurons and the like. Could there be an argument that there are not really any colours? What about an argument that colours do not make sense? What about this:

One photon hits your eye and you might see one of the primary colours. If two photons hit your eye at the same time, then you might see a secondary colour. But only if three arrive together could you see brown. Brown should therefore be the brightest colour. But in fact, it isn't (yellow is). Does that mean that colours do not make sense? Does it mean that there are not really any such things as colours?

Obviously not. Indeed, how could any argument mean that there are no colours, when we can see that there are colours?

And how could any argument mean that there are no collections, when there clearly are things being referred to collectively?

There are, for example, ten words in this final sentence.

Is this Irrational?

Radio 4’s Beyond Belief last Monday was a discussion about embryos (and as far as I can recall it was) between a Scientist (a gradualist, defending the current 14 day rule), a Muslim (who thought that using embryos up to at 40 days, at least, was OK) and an Anglican, say A, who argued that human embryos acquire human rights at the moment of their conception, on the grounds that their humanity (in the relevant sense) is an all or nothing affair (which they certainly have after they’re born, and lack before they exist), whereas embryonic development is indeed gradual (or so I recall—there’s a listen-again feature at the above link, but I’m using my local library’s computers, and being quiet:)
......Towards the end of that 30-minute-programme, A was asked how he would respond to this scenario (more or less:) There’s a fire in a hospital, and A can choose to save either a baby, or else 5,000 embryos (which may have been created for implantation, rather than experimentation), but not both. Naturally A said that he would save the baby, and although my recollection is fuzzy, he began by saying that we don’t always do the right thing, which sounded fudgey. But after posting yesterday’s post, it occurred to me that such a scenario might have yielded a better example of a proposition that was (rationally) regarded as implausible and yet (rationally) believed.
......Incidentally Christopher’s original question explicitly concerned propositions of an ethical nature, as well as the attribution of more physical properties to objects (as in yesterday’s post); and incidentally my first suggestion (in a deleted comment on Christopher’s post) was that we might find it implausible that a certain picture was not beautiful (e.g. because of believing art experts, who say that it is beautiful) whilst simultaneously (and not irrationally) believing that it was not beautiful.
......Anyway, whereas A might know how to save a baby, he might be less sure of what he would be doing with a collection of frozen embryos—e.g. moving them might damage them more than a fire could, for all A knows. And although the scenario might be tweaked to eliminate such uncertainties, note that A had deduced that embryos probably have rights, from what was essentially a lack of information about their (apparently continuous) development as human beings; so there would naturally be more uncertainty in A’s belief that embryos have human rights than in his belief that babies have them. And since the effect of such uncertainty upon a decision naturally increases with the value of what rides upon that decision, hence it would not necessarily be immoral (or irrational) for A to save the baby’s life.
......A could believe quite rationally that saving the baby would be best, because it is only implausible (not obviously false) that saving the baby would (ethically) be better than saving the embryos, hence saving the baby would (ethically) be better than saving the embryos—that is so even if 5,000 people should, all else being equal (tweaking the scenario if necessary), take precedence over one person (this is quite standard, e.g. see the opening paragraph of Hawthone and Stanley's forthcoming Knowledge and Action), but as A pointed out, such Utilitarian assumptions might not be valid (e.g. if each life had infinite worth) and were certainly inappropriate to that radio programme, whose issue was medical scientists using (or killing) embryos, not failing to save them (cf. a doctor choosing to save six people rather than one, which might be reasonable, compared with a doctor killing one person in order to get organs to save the lives of the six, which would be insane:)

What is rational?

The so-called “scientific atheists” claim that while science is rational, religious thinking is irrational (seemingly oblivious to the cogency of, for example, McGrath’s reasoning), but what do they mean by “rational”? Do they mean that our beliefs must cohere with each other and be proportioned according to the evidence? But empirical data has to be interpreted according to an existing system of beliefs, and that system can be modified catastrophically (in the mathematical sense) in the light of new evidence.
......Even within science, scientific paradigms change. And catastrophic modifications must be occurring in the minds of individual scientists as they grow up; presumably they do not fixate irrationally upon the first system of beliefs that they acquired. Indeed, similar modifications are probably occurring for all of us, a lot of the time, and so the coherence of our beliefs would seem to be more of an ideal than an actual possibility.
......Furthermore, in order for our beliefs to be realistic, they should depend upon how the world is, but we judge the importance and likelihood of possible evidence in the light of our existing beliefs, so we are rather at the world's mercy when it comes to being rational. Although I’m not quite sure what “rational” means.

Imagine, for example, that Dawkins is abducted by aliens. Given such quantities of first-hand experiences, Dawkins would probably change his beliefs about UFOs. Had another man come to him earlier with such a story, Dawkins would surely have told him to take seriously the probability that he had imagined the abduction. And when Dawkins returns to Earth, and that empirical data turns into memories of a very strange character, he might naturally and not unreasonably revert to his original skeptical beliefs about UFOs (and regard the abduction as a dream). His friends would no doubt be pleased by his return to sanity.

Now, suppose that McGrath is shown all of the primary data and all of the reasoning that atheists find so important (suppose that he does not already know it), and suppose that that evidence is so good that he loses his faith, temporarily, but that he similarly reverts to his earlier beliefs eventually (and regards his loss of faith to have been the work of the devil). Scientific atheists would hardly say that that reversion was rational.

We can suppose that Dawkins and McGrath were both attempting, at key moments within those fictions, to choose between one set of beliefs (with their explanation of the data) and another set (with their explanation). Making such choices rationally is clearly consistent with being in two minds about things, from time to time. It is consistent with holding inconsistent beliefs, from time to time. And whilst those two examples are naturally rather odd, that was just to keep them simple. People who try to think things through will often find themselves in analogous but much more complicated situations, often with little rational expectation of making their minds up in any reasonable time. But in various situations, one set of beliefs might be more appropriate than another. Is such appropriateness more important than coherence, when it comes to being rational? And could such appropriateness be a big part of what “scientific atheists” mean by science being rational and religious thinking irrational?

What's in a Name?

There is a famous fictional character called Sherlock Holmes. If you discovered that there was a real Victorian detective called Sherlock Holmes who lived in London, then that might be the same person (perhaps the stories were not fictional), but if you met a detective called Sherlock Holmes who lived in London then he would definitely not be the same person, because he would not be fictional. Would it matter whether he was made of 10-dimensional string or not? No. Would it matter if he was made by God, out of whatever God would have made him out of? No, how could it matter what he was made of? Now, if you saw a movie about Sherlock Holmes, then that character could well be the same fictional character, even if the character in the movie was not exactly the same as the character in the books. But what about a character in a video game based on the Sherlock Holmes books? I ask because it is conceivable that we are now living in a Matrix-style world, and it seems to me that it would not matter what the detective that you might meet in the real world would be made of. What if he was like a character in a video game, though?

Dianniversary

Today is the tenth anniversary of Diana’s death (her tenth deathday if you like). According to Gordon Brown, this should be a public holiday (Diana Day). And Francis Wheen, on pages 154-5 of How Mumbo-Jumbo conquered the World, recounts how:

Nine months after the accident, the Mail serialised a book by Rita Rogers, the Derbyshire psychic whose ‘extraordinary powers’ had so impressed Diana and Dodi. She disclosed that at her first meeting with Dodi Fayed the previous summer she immediately had ‘a feeling of danger’: she saw a black Mercedes and a tunnel, and ‘felt there was a connection with France’. Extraordinary indeed: only the most mean-spirited sceptic could have wished for some sort of corroborative evidence, such as a letter from Fayed thanking Rogers for the warning about driving through French tunnels.

Ten years ago I was working nights, so I got the early edition of the Sunday Mirror on the way home, in which Carole Malone wrote that she was sick of hearing about Diana (providing some balance to the supplement, which had lots of pictures of Diana in a bikini on a boat in the tropics). Paradoxically, she must have been very upset by Diana’s death (and not just because it meant that she heard a lot more about her).

Blockquoting

Blogging will be even lighter than usual over the next few days... but incidentally I've been avoiding using the Blockquote button, because the text after the quote gets squished up, but there are apparently two sorts of solutions. One is to insert (under Edit Html) span style="line-height:1.6" (within the usual pointy brackets) after the quote (adding the usual /span at the end); and another, more permanent solution is given here.

An Argument for Atheism

I shall, in this post, take atheism to be the belief that there is (probably) no God, where God will be defined to be an omniscient and omnipotent being that is also totally good (much as Richard defines Him), and I shall argue (as I did briefly in a comment on this post of Richard's) that such a God is (probably) impossible. There are of course other definitions, e.g. atheism is sometimes regarded as the absence of a belief in God, so that it would include both atheism (as defined above) and agnosticism (the absence of a belief either way, which includes the belief that knowledge either way is impossible, which is another definition of agnosticism), and God is sometimes defined to be the Creator of this Universe.
......Such a Creator would know (more or less) all that could be known about this Universe, just as an author would know all about her story, or a painter all about his painting; and similarly, such a Creator would have (more or less) complete power over this Universe. So, it may have been that God was originally defined to be the Creator, and that it was then deduced that such a God would be all knowing and all powerful in that sense (which is, after all, the sense that concerns us, as beings within this Universe), and that such properties only then became definitive, e.g. through their apparent utility—certainly many arguments (as in that linked post) do begin by defining God to be infinitely perfect (rather than the Creator). So, I’ll now argue that such a God, which I’ll refer to as ‘He,’ is (probably) impossible.
......My argument is primarily concerned with omniscience, with God knowing absolutely everything. Not only does He know everything about His Creation (as any Creator would), He also knows whether or not there are, for example, other Gods. If there are any, He knows everything that they know, including precisely what it is like to be them (which might imply that there is only one God), and if not then He knows how he knows that there are not. But it is quite inconceivable how he could know that there are not any other Gods (either at all, or beyond those that He does know all about). The problem is not so much with the “omni,” but with the “science.” It would of course not follow, from some conjectured infinitude having inconceivable properties, that it did not exist; but the concept of knowledge is the concept of true beliefs that are in some way tied down to (or that in some reliable way arise from) the things known about, and we are here considering one being’s knowledge of the non-existence of other similar beings, where there might well be absolutely no connection between them.
......Ironically this problem (for this fairly common kind of theism) resembles a fairly common reply to atheists, who are told that while they might obtain a justified belief that there was a God by His revealing Himself to them, they could hardly obtain scientific knowledge that there was not a God (not even in Heaven) just by failing to have had such a revelation. Imagine (for an analogy) completely separate spacetimes with absolutely no causal connections between them—how could any being, in one of them, know anything about what was going on in the others, or even whether or not there were any others? Similarly, even were there only one spacetime, and a being within it had that true belief, how could that belief be justified?
......By hypothesis God would know that He knew everything, and He would also know how he knew that there were no other Gods (beyond any He might know about more directly, via informative connections), but how could that be? Could He have deduced that fact from His knowledge of His own omniscience? But how could He not then know that such circular justification would not make His belief (that there were no other Gods) knowledge? It is all very well for us to define God to be omniscient, because we can then ask whether or not God exists, but God could hardly do that! In short, the concept of omniscience (in this strict, absolute sense) seems to be self-contradictory. It seems to be, but it may not be, but as there seems to be little logical room for manoeuvre, I regard that conclusion as at least very likely (and not necessarily inconvenient for the theist, as I mentioned here).
......Regarding omnipotence, if God has the power to do absolutely anything, then could He make 2 plus 2 equal 5? If not then it again seems that we cannot interpret His definition in such a strict way after all. And of course, God would not get any less implausible were combinations considered, such as omniscience and (via free will) responsibility, together with omnipotence and (via this Universe existing, whether or not God created it) evil. After all, if God knows what we are going to do, and if He could have stopped us but did not, then, given that He is good, it seems that whatever we do must also be good, or at least (since we are only human) good enough (in what might have to be the best of all possible worlds), which seems unlikely.

Pseudorandom Facts IV-VIII

There might, after all, be something to be said for putting (within this game) all the facts and tags into just one post, so I’m listing my remaining 5 here. First the facts: I prefer boots to trainers, mushrooms to cheese, wasps to spiders, Risk to Chess, and Futurama to the Simpsons. And I'm tagging (for no particular reasons) Alan, Jim, Justin, Leland and Dean.