The Universe of Discourse : There are two kinds of theorems
The Universe of Discourse
Mark Dominus (陶敏修)
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There are two kinds of theorems<br>The road to epsilon-zero: Coin-moving games with no coins<br>Seven books I keep close because I love them<br>The road to epsilon-zero: Infinite Nim as a coin-moving game<br>“Steph Curry: fluke or breakthrough” ten years later<br>The road to epsilon-zero: Nim always ends, even with infinite ordinals<br>The road to epsilon-zero: ordinals as nim-heaps<br>Starting to understand epsilon-zero<br>It's our language now!<br>I owe my life to a 1913 road rage incident<br>Deciphering basmala<br>My 1992 view of the problems of computer programming in 1992
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Tue, 11 Aug 2026
There are two kinds of theorems
In mathematical study there are two kinds of theorems, which serve<br>very different purposes. Math instruction follows the same pattern.<br>Students are often very puzzled by this, and rightly so, because it's<br>never explained, or at least I've never seen it explained. There is<br>this crucial, critical piece of mathematical methodology which is<br>never made explicit, students just have to figure it out on their own,<br>and many of them never do.
When we do mathematics, we construct a simplified model of some<br>phenomenon. For example, Euclidean geometry is a simplified model of<br>how shapes and lines actually work.
In formal geometry, things are simple: lines have no thickness, and<br>three or more lines might all intersect at the exact same point. There<br>are perfect circles, where every point is the exact same distance from<br>the center, and there are perfect rectangles with perfectly straight<br>sides and perfectly equal angles.
Real shapes aren't like this. Nobody can draw an infinitely thin<br>line. Nobody has ever seen a geometrically perfect circle or<br>rectangle. Three lines, however carefully drawn, will always<br>intersect in three different places. That's okay! The point of<br>geometry is to construct a simplified model that is easier to deal<br>with.
When we're setting up a mathamtical model, we start by describing its<br>basic objects, like points and lines, and with axioms and postulates,<br>what properties we intend the objects to have. For example, Euclid has:
A line is breadthless length.
A circle is a plane figure contained by one (curved) line with a<br>point inside, the center, so that the segments from the center to the<br>boundary of the circle are always of equal length
All right angles are equal.
Having done that, we state and prove the theorems of the first kind.<br>We're not studying the actual phenomenon yet. We're not yet trying to<br>learn anything new about shapes and circles. Instead, we're<br>investigating the model itself:
Is the model accurate? Does it seem to lead to wrong conclusions? What<br>happens if we try to prove things that we know are are false? The<br>proofs should fail! If they don't, something is wrong with the<br>model, and we need to fix it.
Is the model powerful enough to prove at least simple things about<br>the actual phenomenon? If it can't encompass the simple aspects of<br>the phenomenon, we haven't much hope of using it to understand more<br>complex aspects.
So for example Euclid starts by proving extremely simple theorems.<br>For example, propositions 4 and 5:
Proposition 4 : If in triangles !!ABC!! and !!A'B'C'!! we have<br>!!AB=A'B'!! and !!AC=A'C'!! and !!\angle A = \angle A'!!, then the two<br>triangles are congruent.
Proposition 5 : If two straight lines cut one another, then they<br>make the vertical angles equal to one another.
(That is, !!\angle CEB = \angle AED!! and !!\angle CEA = \angle BED!!.)
Obviously, yes, anyone can see that! We didn't need to develop a<br>whole mathematical theory in order to discover that vertical angles<br>were equal. Everyone already knew that, long before Euclid. The<br>point of proving this theorem, the real discovery, is: our simple<br>model is strong enough to demonstrate that vertical angles are<br>equal. The theory didn't explicitly include anything about vertical<br>angles, but the vertical alngle theorem was latent in the model<br>anyway.
Consider the opposite situation, where we couldn't prove that<br>vertical angles were equal. Or worse, what if the model allowed us to<br>construct a pair of unequal vertical angles? Would this tell us<br>something about vertical angles?...