- Investigative|Synthetic|General = Operational science
- Investigative|Synthetic|Particular = Historical science
- Non-investigative|Synthetic|General = Philosophy 1st order
- Non-investigative|Analytic|General = Mathematics
- Investigative (empirical) means that the tests are done on the world.
- Non-investigative means that ideas are cognitive and common to man.
- Synthetic means that these ideas are potentially falsifiable based on experience.
- Analytic means that it is not falsifiable based on experience.
- General means that the discovery is a global truth.
- Particular means that a specific event is being described.
And, Non-investigative knowledge cannot be particular. If it is knowledge common to all men, then it is knowledge that is generalisable. This excludes #6 and #8.
We are left with the original 4 categories.
The interesting thing about these categories is that philosophers rate Non-investigative knowledge as more foundational than Investigative knowledge. And if one thinks about this, it makes sense. Investigative knowledge relies on the truth of Non-investigative knowledge. Scientists can see that scientific theory is subservient to mathematics. You cannot say that the theory of gravity is true unless you also hold that the mathematics which is used to describe the theory is also true.
Note that analytic knowledge is not falsifiable. This is because it is derived formally (deductively). One starts with several axioms and, assuming they are true, the rest follows. Mathematical theorems are not accepted true unless every step can be confirmed to be true. Several theories remain unresolved because a mathematician has not solved it. And once it is solved (and confirmed there are no errors) then it cannot subsequently be disproved. 2 + 3 = 5 remains true forever. No new discovery could disprove this.
Note the distinction between operational and historical science (I have previously discussed this). Operational science identifies global truths such as the conservation of energy. This has been well documented, but could potentially be disproved. Historical science will make statements about specific previous events such as when the Polynesians migrated into the Pacific. Further investigation could challenge the accepted norm (or confirm it).
So what of 1st order philosophy? Why should Non-investigative synthetic knowledge take priority?
This is because it is foundational to both science and mathematics. There are several things that man holds true that can only be described as self-evident. They seem true, and most people hold them to be true, but how does one prove them to be true?
Examples include self-identity, and the law of non-contradiction. How does one prove that
- A = A; or
- A ≠ ¬A
So Non-investigative knowledge is the most foundational. 1st order philosophy primarily from whence we get our axioms, and mathematics secondarily as it is deductively certain.
Investigative knowledges are less foundational. Both rely on the Non-investigative knowledges.