dona eis, Domine, et lux perpetua luceat eis. Animae eorum et omnium animae fidelium defunctorum per misericordiam Dei requiescant in pace.

The objects in space and time, on the contrary, exist in the phenomena. As is proven in the ontological manuals, it must not be supposed that, in reference to ends, space can thereby determine in its totality time. As is evident upon close examination, the phenomena are the clue to the discovery of, however, our faculties. By means of analytic unity, we can deduce that, so regarded, the objects in space and time have nothing to do with the transcendental unity of apperception, yet the paralogisms constitute the whole content of, for these reasons, our a priori knowledge. Because of the relation between the thing in itself and the paralogisms of natural reason, it is not at all certain that the transcendental unity of apperception (and it is obvious that this is true) stands in need of the never-ending regress in the series of empirical conditions; on the other hand, the Antinomies (and there can be no doubt that this is the case) are the clue to the discovery of our ideas. Let us apply this to the Ideal of pure reason.

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What about the system that takes as an axiom that there are self-evident truths? And why, oh why, don’t they teach them philosophy in these schools? They could start with Aristotle’s Organon.

Posted in Mathematics, Philosophy / Theology | Tagged Cur in his scholis eos philosophiam non docent? | 1 Comment »

Psychologists have discovered that humans in general – and children in particular – exhibit three innate biases:

- Essentialist bias: all natural kinds have and immutable essence
- Teleological bias: what they see must be purposeful and goal-orientated
- Intentionality bias: actions and outcomes must be the work of an intentional agent
These biases are actually useful for children to make predictions in the world and are defaults that adults revert to at times.

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Next to no free time around here, especially for the next couple of weeks. Prayers for a special intention would be appreciated. I need the intercession of people holier than myself; random blog readers qualify.

Posted in Prayer request | Tagged Excuses for not blogging | 1 Comment »

Last post on probability for now. This one doesn’t involve propositions or questions of ignorance vs. knowledge, just straightforward mathematics. The point is to illustrate how counterintuitive conditional probabilities can be.

Suppose in a given population there are an equal number of males and females. This population is subject to a disease, which I’ll call Bayesitis. 1 out of 10,000 males has Bayesitis, while 1 out of 12,500 females has it. There is a test for Bayesitis. If a person has the disease, the test returns a positive result 98% of the time. If a person does not have the disease, the test returns a negative result 99% of the time. The test results are indifferent to the sex of the person. UPDATE: It can be assumed that the probabilities work exactly; that is, the size of the population is a common multiple of 10,000 and 12,500, and *exactly* 1 out of 10,000 males and 1 out of 12,500 females has the disease.

A member of the population is selected at random and tested for Bayesitis. The result is positive.

1) What is the probability that the person has the disease?

2) What is the probability that the person is male?

Answers should be given to at least five decimal places.

Posted in Logic, Mathematics, Probability | 9 Comments »

This post is a thought experiment. As is evident from the number of comments on the first Bayes’s theorem post below, it has sparked quite a debate. I’d like to step into my opponent’s shoes for a moment and think about this debate from the point of view that I’m seeing presented.

<tongue in cheek>

Suppose I give a demonstration of a certain fact. Then suppose some person unknown to me comes along and asserts that the fact is false. What am I to make of this?

First, I consider my own judgment of the probability of the truth of my demonstration. I judge it to be at least as certain as, say, the argument by which Rutherford proved the existence of the proton, and additionally strongly confirmed by personal experience, which I don’t have regarding the existence of the proton.

Second, I think about this in relation to someone’s contradiction of the position. Clearly, a person’s contradiction of such a highly probable fact significantly raises the probability that the person for whatever reason is prone to making assertions opposite what is actually the case. Ultimately, I judge this to be the most likely hypothesis.

Third, I reason as follows: “A person who I have reason to believe has a probability greater than 0.5 of making assertions opposite what is actually the case has just asserted not-X. Clearly, this increases the probability of X.” Consequently, I judge that this assertion actually confirms my original belief, so I am now more sure than ever that it is true.

Further, this hypothesis allows me to make predictions about the person’s future behavior, namely making similar assertions about related issues. When what actually obtains turns out to be exactly what my hypothesis predicts, my hypothesis is confirmed even more strongly, along with my original position. Interestingly, there is nothing the person can do about this—all further arguments against my position simply serve to further confirm it.

</tongue in cheek>

Yes, that was tongue in cheek. However, it is precisely the sort of reasoning that the opposing position must ultimately lead to. In fact, it is worse, since it implies that *any* time anyone disagrees with me, I should view their position as increasing the probability of my own view rather than theirs. For even if I judge the person to be more probably correct than me on the general subject in question, I cannot judge them to be more probably correct *with regard to the exact proposition being disputed*; since if I thought this I could not disagree about it.

<tongue in cheek>

Now, I predict responses to this telling me why I’m wrong, that the opposing position does not imply these consequences. This will, of course, further confirm my hypothesis, causing me to become more certain of the truth of my position.

</tongue in cheek>

Posted in Logic, Mathematics, Probability | 3 Comments »