Re: mathematics is not the language of reality

On 2008-03-18, in sci.logic, Rupert wrote:
I would be interested in an explanation of Carroll's Paradox.

Achilles had overtaken the Tortoise, and had seated himself
comfortably on its back. "So you've got to the end of our
race-course?" said the Tortoise. "Even though it DOES consist of an
infinite series of distances? I thought some wiseacre or other had
proved that the thing couldn't be done?" "It CAN be done," said
Achilles. "It HAS been done! Solvitur ambulando. You see the
distances were constantly DIMINISHING; and so -" "But if they had
been constantly INCREASING?" the Tortoise interrupted. "How then?"
"Then I shouldn't be here," Achilles modestly replied; "and YOU would
have got several times round the world, by this time!" "You flatter
me - FLATTEN, I mean," said the Tortoise; "for you ARE a heavy
weight, and NO mistake! Well now, would you like to hear of a
race-course, that most people fancy they can get to the end of in two
or three steps, while it REALLY consists of an infinite number of
distances, each one longer than the previous one?" "Very much
indeed!" said the Grecian warrior, as he drew from his helmet (few
Grecian warriors possessed POCKETS in those days) an enormous
note-book and pencil. "Proceed! And speak SLOWLY, please! SHORTHAND
isn't invented yet!" "That beautiful First Proposition by Euclid!"
the Tortoise murmured dreamily. "You admire Euclid?" "Passionately!
So far, at least, as one CAN admire a treatise that won't be
published for some centuries to come!"

"Well, now, let's take a little bit of the argument in that First
Proposition - just TWO steps, and the conclusion drawn from
them. Kindly enter them in your note-book. And in order to refer to
them conveniently, let's call them A, B, and Z: -

(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to
the same.
(Z) The two sides of this Triangle are equal to each other.


Readers of Euclid will grant, I suppose, that Z follows logically
from A and B, so that any one who accepts A and B as true, MUST
accept Z as true?" "Undoubtedly! The youngest child in a High School
- as soon as High Schools are invented, which will not be till some
two thousand years later - will grant THAT." "And if some reader had
NOT yet accepted A and B as true, he might still accept the SEQUENCE
as a VALID one, I suppose?" "No doubt such a reader might exist. He
might say, 'I accept as true the Hypothetical Proposition that, if A
and B be true, Z must be true; but I DON'T accept A and B as true.'
Such a reader would do wisely in abandoning Euclid, and taking to
football." "And might there not ALSO be some reader who would say 'I
accept A and B as true, but I DON'T accept the Hypothetical'?"
"Certainly there might. HE, also, had better take to football." "And
NEITHER of these readers," the Tortoise continued, "is AS YET under
any logical necessity to accept Z as true?" "Quite so," Achilles
assented. "Well, now, I want you to consider ME as a reader of the
SECOND kind, and to force me, logically, to accept Z as true."

"A tortoise playing football would be -" Achilles was beginning. "-
an anomaly, of course," the Tortoise hastily interrupted. "Don't
wander from the point. Let's have Z first, and football afterwards!"
"I'm to force you to accept Z, am I?" Achilles said musingly. "And
your present position is that you accept A and B, but you DON'T
accept the Hypothetical -" "Let's call it C," said the Tortoise. "-
but you DON'T accept

(C) If A and B are true, Z must be true."


"That is my present position," said the Tortoise. "Then I must ask
you to accept C." "I'll do so," said the Tortoise, "as soon as
you've entered it in that notebook of yours. What else have you got
in it?" "Only a few memoranda," said Achilles, nervously fluttering
the leaves: "a few memoranda of - of the battles in which I have
distinguished myself!" "Plenty of blank leaves, I see!" the Tortoise
cheerily remarked. "We shall need them ALL!" (Achilles shuddered.)
"Now write as I dictate: -

(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to
the same.
(C) If A and B are true, Z must be true.
(Z) The two sides of this Triangle are equal to each other.


"You should call it D, not Z," said Achilles. "It comes NEXT to the
other three. If you accept A and B and C, you MUST accept Z." "And
why must I?" "Because it follows LOGICALLY from them. If A and B and
C are true, Z MUST be true. You can't dispute THAT, I imagine?"

"If A and B and C are true, Z MUST be true," the Tortoise
thoughtfully repeated. "That's ANOTHER Hypothetical, isn't it? And,
if I failed to see its truth, I might accept A and B and C, and STILL
not accept Z, mightn't I?" "You might," the candid hero admitted;
"though such obtuseness would certainly be phenomenal. Still, the
event is POSSIBLE. So I must ask you to grant ONE more Hypothetical."
"Very good, I'm quite willing to grant it, as soon as you've written
it down. We will call it

(D) If A and B and C are true, Z must be true.

Have you entered that in your note-book?" "I HAVE!" Achilles
joyfully exclaimed, as he ran the pencil into its sheath. "And at
last we've got to the end of this ideal race-course! Now that you
accept A and B and C and D, OF COURSE you accept Z." "Do I?" said
the Tortoise innocently. "Let's make that quite clear. I accept A and
B and C and D. Suppose I STILL refused to accept Z?" "Then Logic
would take you by the throat, and FORCE you to do it!" Achilles
triumphantly replied. "Logic would tell you, 'You can't help
yourself. Now that you've accepted A and B and C and D, you MUST
accept Z.' So you've no choice, you see." "Whatever LOGIC is good
enough to tell me is worth WRITING DOWN," said the Tortoise. "So
enter it in your book, please. We will call it

(E) If A and B and C and D are true, Z must be true.

Until I've granted THAT, of course I needn't grant Z. So it's quite a
NECESSARY step, you see?" "I see," said Achilles; and there was a
touch of sadness in his tone.

Here the narrator, having pressing business at the Bank, was obliged
to leave the happy pair, and did not again pass the spot until some
months afterwards. When he did so, Achilles was still seated on the
back of the much-enduring Tortoise, and was writing in his notebook,
which appeared to be nearly full. The Tortoise was saying, "Have you
got that last step written down? Unless I've lost count, that makes a
thousand and one. There are several millions more to come. And WOULD
you mind, as a personal favour, considering what a lot of instruction
this colloquy of ours will provide for the Logicians of the
Nineteenth Century - WOULD you mind adopting a pun that my cousin the
Mock-Turtle will then make, and allowing yourself to be renamed
TAUGHT-US?" "As you please," replied the weary warrior, in the
hollow tones of despair, as he buried his face in his
hands. "Provided that YOU, for YOUR part, will adopt a pun the
Mock-Turtle never made, and allow yourself to be re-named A
KILL-EASE!"

Alas, an explanation is beyond my measly skills.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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