On Page 21 Section 8 of the English translation, there is lengthy geometric explanation. I'm having difficulty picturing what is being explained due to wording and perhaps loss during translation.
Can anyone help explain the geometric circle example that is given and it's relation to $(A \equiv B)$? I'm uncertain if point B should be the center of the circle or on the circumference.
Also, if $\vdash (A \implies X) \land (B \implies Z)$ were defined how would $\equiv$ identity of content resolve?
Geometric Example:
Thank you!
First, here's the geometric example illustrated:
Now imagine that the ray “revolves around $A$”. That is, it changes direction while still passing through $A$:
So, as the ray turns continuously, point $B$ moves continuously along the circumference of the circle.
(This is geometrically a little bit confused, but let's agree to understand the geometry the way Frege asks us to: even when the ray is perpendicular to the diameter of the circle, there are still two intersections, one defined by $A$ and the other defined by its “other” intersection with the circumference. It just so happens that the two intersections are in the same place.)
Now here's why Frege introduces this example. As mathematics has evolved, so has our understanding of the “=” symbol. It is much more philosophically complex than it appears. What do we actually mean when we say that $$2+2=4?$$ The accepted explanation prior to Frege was that the two sides are just names for the same entity, analogous to the way one would write “Mark Twain is Samuel Clemens”: they are two names for the same individual, and the “=” sign means that the two entities are the same.
Frege wants us to understand that this account is insufficient. If “$2+2”$ and “$4$” are simply different names for the same object, and if “=” simply means that the objects are the same, then the meaning of “2+2=4” should be the same as the meaning of “4=4”. Which, of course it isn't. To consider a more extreme example, take Euler's famous formula $$e^{ix} = \cos x + i \sin x.$$ If the left and right sides are simply names for the same thing, why is this formula more interesting than the less-famous formula “$e^{ix} = e^{ix}$”? Why do we think that “Mark Twain is Samuel Clemens” is more interesting than “Mark Twain is Mark Twain”, if both assertions mean exactly the same thing?
In Frege's example, we have fixed point $A$ and varying point $B$. When the ray is perpendicular to the diameter, $A=B$. Are $A$ and $B$ simply names for the same point?
Frege wants to distinguish two kinds of meaning, which he calls “sense” (Sinn) and “reference” (Bedeutung). In Frege's view, “$2+2=4$” has content different from “$4=4$” because the sense of “2+2” is different from the sense of “4”, even though the thing to which they refer (the number four) is the same in both cases.
Similarly, at a particular moment, “the fixed point $A$” and “The other intersection of the ray with the circumference” may happen to refer to the same point, but their sense is still quite different.
“2+2”, says Frege, is not merely a different way of writing “4”.
The sense of “2+2” is the result of an arithmetic addition process. The sense of “4” is not. They are different ways of determining the content. To assert that “2+2=4” is to assert that these two different ways of determining content do determine the same thing, in this case the number four.
Now what does Frege want us to understand by $\vdash A\equiv B$? There the language is a little mixed. I take $A$ and $B$ to be names for signs. For example, $A$ might be the sign “2+2” and $B$ might be “4”. (Or $A$ might be “the fixed point $A$” and B might be “the other intersection of the ray with the circumference when the ray is perpendicular to the diameter”.) Then by $$2+2\equiv 4$$ Frege means that the signs “$2+2$” and “$4$” refer to the same object. And $$\vdash 2+2\equiv 4$$ means that we judge or assert that this is actually the case.