Kant's Fingers

A brief follow up to the previous allusion to Kant...

In the Critique of Pure Reason, Kant defined the analytic as containing a non-additive predicate, while the synthetic as having an additive predicate. While I still need to work out the details, it's clear that Kant also implicates the digital and the analog with this famous distinction. As I define them, "digital" has a special relationship with analysis, while "analog" with synthesis. The terms are so similar that, in some instances, they act as synonyms (digital=analysis, analog=synthesis). I'm sure Kantians would scoff at the attempt, but I'm keen on misreading Kant here, portraying Kant's analytic as digital, and his synthetic as analog. Again, I need more time to see where this goes, but at the very least it offers a new perspective on Kant's famous expression "7 + 5 = 12" and his somewhat controversial notion that such mathematical expressions are, in his terms, "synthetic." (Critique of Pure Reason, B15). Is math also an "analog" technology for Kant, and if so how? 

Why does Kant use "synthetic" to describe mathematical expressions like "7 + 5 = 12"? The key is to see the expression as a sequential calculation rather than an arithmetical expression (more like algebra). Kant treats the expression like a sentence with a subject and a predicate rather than an algebraic equation (which is reversible). Running left-to-right, the granular terms "7," "+," and "5" are synthesized into a single term, "12." The predicate is literally additive, even if it appears that terms have been removed (from three down to one). Hence the *right* side of the expression ("12") is synthetic, not the left side or the expression as a whole.

In this part of the Critique, I've always found one sentence particularly jarring. Kant claims one must "seek assistance" by appealing to "one's five fingers." Appeal to the fingers?! There are two problems with this sentence. First, the a priori means *not* appealing to experience, yet here, in a seemingly off-hand comment, Kant appeals to experience. Second, you can't explain number by saying "look at my fingers." That's just using digits to explain digits. A finger is just a different kind of dogmatic foundation for number; it doesn't "ground" arithmetic. How can Kant claim to define pure mathematics if he absolutizes digitality via biological anatomy? This sentence is very strange, and it seems easier just to omit it.

No matter, let's consider a few other issues that follow from Kant's discussion of "7 + 5 = 12." The first is about reversibility. As a simple formula, "7 + 5 = 12" is interchangeable with "12 = 7 + 5." But, as a Kantian sentence, the expression is not reversible. The left side, "7 + 5," is not synthetic in nature, but instead carries greater analytical value. We have, in a sense, analyzed (i.e. digitized) the number 12 by breaking it down into two terms plus an operation.

Interestingly the expression also destroys information when read left to right, and hence is a form of compression. The expression goes from three terms to one (from two values plus an operator down to one value). The "atomic" information on the left side of the expression is lost. If I give you "12" you can't tell me which unique pair of integers were summed to make it; it could be 10 + 2, 8 + 4, etc. That information is lost. In this sense, the left side of Kant's expression is "more" analytical and hence more digital, while the right side, as synthesis, is more analog. When read left to right the expression "7 + 5 = 12" is, quite literally, a digital-to-analog conversion.

At the same time Kant claims that all mathematics is synthetic, not just "7 + 5 = 12." So, again, I'd need to think more about this, but my intuition is that the first Critique will emerge as a key text for digital-analog theory, once the terms are effectively glossed. Perhaps someone else will take up the question.

(Regarding reversibility and compression, electrical engineers deal with similar issue in the design of logic gates. For example, a gate with two inputs but one output effectively "compresses" information from two bits to one bit. Such gates are also not reversible, since you can't work backward from one bit and reconstruct the two bits that made it. This has dramatic consequences for digital physics, since physics requires reversibility and conservation of information; Fredkin's reversible logic gate was invented to address these kinds of problems.)