Wednesday, November 16, 2011

The fecundity of Spinoza's claims

Say that the fecundity of a claim in a logically interconnected text, like Spinoza's Ethics, is the number of claims that logically depend on it. Using the Tredwell adjacency data, I sorted the claims in Spinoza's Ethics in order of decreasing fecundity. We can measure the fecundity in percentages: the percentage of the claims that depend on the given claim.

The result is here. (The explanations of what the items are are here, from Tredwell.) Fecundity is a measure of how fundamental a given claim is to the system.

The twelve most fecund claims, with their number of dependants, are:

  1. 1A04: 300 (77.3%)
  2. 1D03: 296 (76.3%)
  3. 1D04: 295 (76.0%)
  4. 1D05: 292 (75.3%)
  5. 1A06: 291 (75.0%)
  6. 1A01: 291 (75.0%)
  7. 1P01: 290 (74.7%)
  8. 1A05: 290 (74.7%)
  9. 1P04: 290 (74.7%)
  10. 1P02: 289 (74.5%)
  11. 1P03: 289 (74.5%)
  12. 1P05: 289 (74.5%)
Unsurprisingly, they're all from Part I of the Ethics, and unsurprisingly the first six are all axioms or definitions.

The most fecund is Axiom 4, that the cognitio (understanding?) of the effect depends on the cognitio of the cause, which, through Spinoza's overreading of it (it sounds like a weak claim, and that's why we are tempted to agree, but in fact it is a strong claim), becomes the root of the epistemologically central 2 Proposition 7, which says that the order and connection of things is the order and connection of ideas. In fact, it is largely through this 2P07 that Axiom 4 gets its fecundity: 2P07 has a fecundity of 60%, and assumes nothing other than 1A04.

The most fecund derived claim is 1 Proposition 4, that distinct things must differ in attribute or mode.

Unsurprisingly, the ontological argument is central: the fecundity of 1 Proposition 11, that God exists, is 73%.

The most fecund claim from outside of Part 1 is the aforementioned 2 Proposition 7, whose centrality cannot be denied.

There are 103 propositions that have zero fecundity.

Axiom 2 of Part I has zero fecundity in the database I am using, as do 5A02 and 2A08. Due to the limitations of my method, axioms and definitions with zero fecundity don't appear in the results I linked to, though I may fix that eventually. The case of Axiom 2 of Part I interesting and surprising, since it basically states Spinoza's version of the Principle of Sufficient Reason. My feeling is that it is implicitly used all over the place.

The least fecund axiom that actually gets used is 5A01, about contrary actions, which has only one dependent. The next, somewhat more fecund axiom is 4A01, at 4% fecundity, which says that for any thing, there is a stronger thing which can destroy it.

Surprisingly to me, the least fecund axiom from Part 1 is 1A03, at 8%, which basically affirms that causation is deterministic. This may initially suggest that Spinoza's causal determinism is not as central to his thought as it is normally thought to be. But that might be too quick, because I suspect that much if not all of the deterministic import of 1A03 is found in 1A04, especially as interpreted by 2P07 and with the understanding the the logical connections between ideas are always entailment relations.


some kant said...

This is pretty cool. I guess you could also do the reverse: find the "most dependent" propositions.

Alexander R Pruss said...

Here you go.

Most dependent, i.e., richest ancestor tree:
4P46 at 46.1% "He who lives according to the guidance of reason strives as much as possible to repay hatred, anger, or contempt of other with love or generosity."

This one ultimately depends on: 1A01 1A04 1A05 1A06 1A07 1D01 1D02 1D03 1D04 1D05 1D06 1D08 2A01 2A02 2A03 2A04 2A05 2A06 2A07 2D02 2D12 2O01 2O03 2O04 2O05 2O06 3D01 3D02 3D03 3E01 3E07 3E23 3O01 4D01 4D02 4D08, as well as various intermediate steps.

Second-most dependent:
4P59 at 45.6%: "To all actions to which we are determined by an emotion wherein the mind is passive we may, without the emotion, be determined by reason"

These suggest that "Ethics" is a good name for the book--the arguments really do culminate in the ethical propositions, and these ethical propositions draw on a lot of controversial stuff.

James said...

Fecundity seems like a filter, doesn't it?