Foundations of Mathematics

I watched this interesting 2017 PBS video "Crisis in the Foundation of Mathematics" today. I was struck by how in the Zermelo-Fraenkel set theory of the foundations of mathematics, the existence of infinite series has to be defined as an axiom: it's not automatic that they exist.

Another thing that came to mind is the weird way that mathematicians try to formulate a foundation for their discipline, for example, in order to broaden the natural (counting) numbers into the integers (that is, to include the negative numbers), they define an integer as the difference between two natural numbers. What I don't understand is why they don't just define negative numbers as an abstraction that incorporates the "take away" operation (subtraction) into the number. Imaginary numbers would then be a kind of "half-way" take-away, what amounts to a ninety-degree rotation (half of 180°).

I say "I don't understand," but seriously: why should anyone expect me to understand? My training is in physics, not mathematics, after all.

Death to Self and Discernment

That outward circumstances play a part in the formation of a spiritual judgment may be seen by merely looking at the kind of circumstances that interior souls at one time or another have to face. It is easy enough to estimate the effect of these things upon their characters. Loss of material goods conduces to a man's discernment. With detachment from outward standards comes a greater reliance upon the significance of the inward.

Sickness conduces to discernment. There is nothing like a long illness to teach a man the difference between true and false compassion. If only from the sight of his own self-pity, he learns the value of entering into the pains of others.

Suffering of every kind—and especially the suffering of temptation—fosters the potentiality of discernment. Not only is the genuine need distinguished from the sham, but even in the need that is unjustified, that is brought upon itself, an element of sincerity can be discovered that demands an act of understanding.

Solitude ministers to discernment. Sometimes it is born in it. In fact, one wonders how a soul can come to possess the discerning spirit without the help of solitude and silence.

And, above all, in prayer: in the practice of unrelenting prayer, hours of it and carried on over the years, a soul chiefly learns to judge according to the spirit. Discernment is nothing other than this: the power to interpret God. How, short of the directly miraculous, can God's will be interpreted as it is capable of being interpreted apart from the light of prayer?

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Dom Hubert van Zeller, How to Find God ... and Discover Your True Self in the Process (Manchester, New Hampshire: Sophia Institute Press, 1998), 206.