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[Errata: please replace the word "intuition", wherever it exists in
capitalized form in this document, by the word "G00035" -- dbl]

[Disclaimer: some the material below is not actually implemented in the
existing AM program -- dbl]

The Intuitions facet of a concept C, abbreviated C.Intu, contains a series
of helpful little programs for viewing C in an abstract way. There are only
a few global scenarios, such as firing arrows, and any concept which can be forced
into that scenario somehow has a corresponding entry on his Intutions facet,
explaining how to do this. 
For example, one intuition for a function is to fire an arrow from each element of
the domain set somewhere into the range set.
For a relation, any number of arrows may be fired; for a bijection, each
target in the range set must be hit. For a set, the intuition can include the
set of archers, of targets, of arrows, etc.

In addition to amusing the human user, what good are these images?
As with hman intution, the Intuitions parts can be used to suggest
-- and to convincingly justify -- conjectures. 

The intuitions might indicate that
that if a surjection exists from A to B, and from B to A, then each
archer in A will fire an arrow and every target in B will be hit. The
archer standing by that B-target can then re-fire the arrow 
(if he gets more than one, he ignores the extras)
according to the
second surjection, and hit every target in A.
But each archer only fired one arrow, so the number of returning arrows
must equal the number sent out initially. So no B-archer ignored any arrow.
So no B-archer received mor than one. So the original surjection was a bijection.
This is the famous Schroeder-Bernstein theorem.
AM is in  position not only to propose it, but to sketch a proof!

Each Intuition entry is like a  "way in" to one of the few global scenarios.
It can be characterized as follows:

.BN


λλ One of the sailent features of these entries -- and of the scenarios -- is that
AM is absolutely forbidden to look inside them, to try to analyze them. They
are ⊗4↓_opaque_↓⊗*. The intuition for Cross-product might contain an algorithm for
multiplication, and it would be pointless to say that AM discovered TIMES
if it had access to that algorithm all along.

λλ The second characteristic of an Intuition is that it be ⊗4↓_fallible_↓⊗*.
As with human intuition, there is no guarantee that what is suggested will be
verified even empirically, let alone formally.

λλ Nevertheless, the intuitions are very ⊗4↓_suggestive_↓⊗*. Many conjectures can
be proposed only via them. Some analogies (see the next subsection) can also
be suggested only via common intuitions.

.E

It is somehow necessary -- and yet unfair -- to allow intuitions to have 
the power to magically sugget otherwise obscure potential relationships.
With humans, it is "fair" to draw upon intutions, since any new advance is a
new-to-Mankind advance. With AM, working at such a simple level, it is dangerous
to provide intuitions, since they may contain the "secrets" of already-well-known
mathematics in them.
On the other hand, if a small set of intutitions were sufficient to motivate a
huge chunk of mathematics, then those intuitions themselves would be worth
writing about, teaching to kids, etc.

We shall cover one more simple example, and then drop this incompletely-implemented
idea.  I think there is some real opportunity for research here; it is what you
get$$ Notice the careful wording here. $ for reading the thesis in this much detail!


The relation "α>" (x is greater-than y)  has the following intuition entry:

.B816

SEESAW: Left-persons←x; Right-person←y; Final-tilt=Left.

.E

There exists an opaqe function named SEESAW, which rpresents the scenario of
people on a seew-saw. It takes various arguments, and produces various results.
In this case, some the arguments are not specified (e.g., the initial way the
board is tilting). The entry for α>.Intu says to return True iff the final
position of the board is tilting left; i.e, if the heavier person is x.
Since the seesaw intuition contains the notion of antisymmetry
(reversing the arguments reverses the result), it might cause AM to consider the
conjecture that xα>y iff NOT(yα>x). This fails for extreme cases (x=y) but is
otherwise true.
Also, the intuition for α< contains a reference to SEESAW. SO AM would think that
they might be related, and intutitively since the final position is always either
left or right, the conjeccture would get proposed that xα<y or xα>y. Again this
fails in an easily patchable-way. It leads to what is known as the axiom of 
trichotomy.

Perhaps that is the best possible use of intuition: to derive and justify what
are considered as axioms, as primitives in a formal sense, not "explainable" in
any other manner.

Another use for intuitions is to try to verify conjectures. For exaple, say the
conjecture that Doubling a number always gives an even number has been proposed
for empiricial considerations. It would be nice to intuitively grasp that
relationship. If it cannot be verified, that increases its strangeness$$
Such failure increases the chance that the conjectuere 
might be false, or that if it ⊗4is⊗*
true, it is probably either very deep or totally useless. $.

Without intuitions like these, AM might not get trapped into believing in the
necessity of the parallel postulate; so they are a mixed blessing.