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Sunday, February 19, 2017


It's no secret that Trump hates the press (and the press hates him right back). Continuing his adversarial relationship with the press, tonight at his Florida rally, Trump said:
"Nothing can be believed which is seen in a newspaper. Truth itself becomes suspicious by being put into that polluted vehicle."
Fake news and all that.

Wait! What?

He was quoting Thomas Jefferson written in a correspondence on June 14, 1807!


I guess Presidents and the press have had an adversarial relationship for a really, really long time!

Saturday, February 18, 2017

Fun With Infinity

Infinity and infinite series and sets are concepts that stretch human intuition to the breaking point and as a result, are kinda fun - for masochists. The particular infinite series I'm gonna look at today is:

S = 1 + 2 + 3 + 4 + ...

What is the value of S?

The NY Times recently had an article demonstrating that a possible answer is -1/12 (there's a more rigorous proof that shows the answer is indeed -1/12 but is beyond what I can show on a blog). I know that some of you studiously avoid the NY Times and therefore might not have seen it, so I'll duplicate it here with a little more explanation.

There's only one somewhat non-intuitive bit to the proof, so let me address that before I get started. The best illustration of this bit of non-intuition is called Hilbert's Paradox of the Grand Hotel:
Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms... 
Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room.
What this demonstrates is that if I have two infinite sets (such as rooms and guests) with a one-to-one correspondence between each pair of elements in the sets, I can shift over all of the elements of one of the infinite sets, leaving one element without a corresponding element in the other set (room 1 in the example above), yet have all of the other elements of both sets still have a one-to-one correspondence.

Okay, we need to find the values of some infinite series. The first one is:

S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...

What is the value of S1? To find an answer, we add it to itself and do the hotel room operation above (in other words, shift over one copy of the series).

2 * S1 = S1 + S1 =  1 - 1 + 1 - 1 + 1 - 1 + ...
                  +     1 - 1 + 1 - 1 + 1 - ...
                  = 1 - 0 + 0 - 0 + 0 - 0 + ...

or, 2 * S1 = 1
therefore, S1 = 1/2

The second series we need is:

S2 = 1 - 2 + 3 - 4 + 5 - 6 ...

And we start with the same infinite shift operation that we used on the previous series:

2 * S2 = S2 + S2 =  1 - 2 + 3 - 4 + 5 - 6 + ...
                  +     1 - 2 + 3 - 4 + 5 - ...
                  = 1 - 1 + 1 - 1 + 1 - 1 + ...

But that's the same as the 1st series that we already know an answer to:

2 * S2 = S1 = 1/2
therefore, S2 = 1/4

So now let's work on our original series. We'll subtract S2 to help us find an answer:

S - S2 =  1 + 2 + 3 + 4 + 5 + 6 + ...
        -[1 - 2 + 3 - 4 + 5 - 6 + ... ]
        = 0 + 4 + 0 + 8 + 0 +12 ..
        = 4 * [ 1 + 2 + 3 + ... ]

The right hand side is now 4 * S so rewriting we have:

S - S2 = 4 * S

or (subtracting S from both sides)

- S2 = 3 * S

Since we know S2 = 1/4, we have

- 1/4 = 3 * S


S = -1/12


1 + 2 + 3 + 4 + ... = -1/12

This sort of proof, where the sum of an ever increasing series is a negative fraction, makes some people's heads explode. I hope you're not one of them. It's just a little fun with infinity!

Tuesday, February 07, 2017

Bug or Feature?

Congrats to Betsy DeVos, confirmed as the Secretary of Education by literally the narrowest margin possible (Vice-President Pence had to cast the tie-breaking vote in the Senate).

One of the charges leveled against her was that she's completely unqualified to be the Secretary of Education and utterly clueless about what it takes to keep the education bureaucracy afloat.

I can't say I disagree. But is that a bug or a feature?

To me it seems like the entire education edifice is in catastrophically poor condition with kids not being very well educated and/or prepared for life as an adult even though funding has hugely increased over the last few decades. Perhaps a truly incompetent secretary of education will damage the system enough that it simply collapses and then it can be rebuilt from scratch. Especially with online and other tools improving at a rapid rate, catastrophic destruction of the whole thing may be the best way to ultimately improve it.

So, as I say, congrats, but I'm not sure if I wish her good luck or bad luck. A little incompetence coupled with some bad luck may be just what we need right now!

Saturday, February 04, 2017

Hypocrisy on Parade

The NYT runs an intermittent series under the heading of The Stone; it purports to be "a forum for contemporary philosophers and other thinkers on issues both timely and timeless."

I have previously (here and here) rubbished articles for grievously offending my logical sensibilities. Unfortunately, the comments threads were of no help in deciding whether the deficiency was mine or some contemporary philosophers and other thinkers.

Once again, it is time to reach for the Rubbisher.

Peter Singer is something of an enfant terrible: his niche in philosophy is to take a seemingly reasonable position, and extrapolate it to where shock and opprobrium is sure to follow.

Here are some examples:

Abortion: In Practical Ethics, Singer argues in favour of abortion rights on the grounds that fetuses are neither rational nor self-aware, and can therefore hold no preferences. As a result, he argues that the preference of a mother to have an abortion automatically takes precedence. In sum, Singer argues that a fetus lacks personhood.

Similar to his argument for abortion, Singer argues that newborns lack the essential characteristics of personhood—"rationality, autonomy, and self-consciousness"—and therefore "killing a newborn baby is never equivalent to killing a person, that is, a being who wants to go on living."

Speciesism: Speciesism is an attitude of bias against a being because of the species to which it belongs. Typically, humans show speciesism when they give less weight to the interests of nonhuman animals than they give to the similar interests of human beings.

[On the basis that a being able to think of itself as existing over time], one might argue that to kill a normal human being who wants to go on living is more seriously wrong than killing a nonhuman animal. Whether this claim is or is not sound, it is not speciesist. But given that some human beings – most obviously, those with profound intellectual impairment – lack this capacity, or have it to a lower degree than some nonhuman animals, it would be speciesist to claim that it is always more seriously wrong to kill a member of the species Homo sapiens than it is to kill a nonhuman animal.

Altruism: A minimally acceptable ethical life involves using a substantial part of one's spare resources to make the world a better place.

These positions run the gamut from the apparently awful to the seemingly benign. I think they each rest on at least some flim-flammery, by either ignoring inescapable elements of reality — time, say — question begging, or failing to take an argument to where it demands being taken.

But no matter, that isn't what had me casting about for my Rubbisher.

It has come to some degree of notice that Peter Singer is spending significant resources caring for his Alzheimer's crippled mother. For most of us, more or less unburdened by a surfeit of philosophical posing, uhh, thinking, this is a no brainer. However, for Singer, this is clearly verboten, whether on the grounds of altruism or speciesism, at the very least.

Yet, despite his admonitions to the rest of us, he does so, nonetheless.

The philosopher Peter Singer was once attacked for contradicting himself. Singer advanced an ethical theory in which the most worthwhile thing was complex conscious life and feeling, and did not shy away from the logical consequence that the life of a severely mentally impaired human was worth less than that of a chicken. Journalists then discovered that Singer’s mother had Alzheimer’s and that he chose to spend his money taking care of her rather than helping chickens.

They called Singer a hypocrite and The New Republic even ran a cover with a picture of an addled old woman with a walker and the headline “Other People’s Mothers.”

Failing to notice the answer on offer, the author, by definition an esteemed contemporary philosopher or other thinker on issues both timely and timeless goes straight to missing the screamingly obvious:

So, how bad is contradicting yourself?

In philosophy, since Socrates (a troll before there ever was an internet), the answer has been “very bad.” If you find you believe two inconsistent propositions you need to do something about it. You owe a theory.

No, Eric Kaplan, this isn't contradicting yourself, this is allowing yourself that which you prohibit others. There's a fancy word for it, often improperly used, but not here: hypocrisy. Contradiction, entirely unrelated, involves having taken a position, subsequently taken on board discordant information, then reversing, or significantly changing your position; not just for yourself, but for everyone else, too.

Peter Singer has done nothing of the kind. But let's let that slide, so that Kaplan can have his say:

Part of the reason this mother/chicken puzzle is so hard is it runs up against two contradictory beliefs we have about human beings:

a) Humans are meaningful; the things they do make sense

b) Humans are things with causes like anything else — as meaningless as forest fires.

I could burden you with further pull quotes, but I won't because the chase that needs cutting to is right here. Kaplan, and on his behalf, Singer, have skipped right over a fatal error.

What do you think it is? Hint: it is contained in a single word.