Calculus fails on discontinuous functions. Discontinuities exists in our measurements of reality. Therefore calculus is fallible when applied to reality. QED.
Thankfully, discontinuities aren’t all that common when you consider all the shit that does work according to our calculations.
Ask a logician about circular logic and they will tell you it is a fallacy. But, to me, this is an oversimplification. It holds as a fallacy when used in isolation, but when there are feedback loops, circular logic can indeed be used to produce matter from the illusions created by circular logic, or circular reasoning as Wikipedia calls it. Of course, the logicians are right in that circular logic is illusory, but that does not make it useless. Math has imaginary numbers already, why not formalize imaginary logical structures? Society has been running off half baked theories that are consistently revised since we started chasing our crazy half baked ideas. And it has clearly led to great advances. But, knowing that today’s brilliant idea will be tomorrows useful, but discarded broken theory, shouldn’t we attempt to understand the phenomenon? There are small errors in everything, and engineers design by just allowing enough tolerance. But knowing how much tolerance is needed is considered an art, and many would argue it can’t really be a hard science. If you look at Gödel’s work with the glasses of illusions as false, they have a point. Any theory consistent with existing theories will not “complete” (finish, as in “we’ve solved it, there are no more theories”) all the theories, as long as the set of theories is past a certain level of complexity. And that level of complexity exists everywhere. Out best hope is to predict as much as we can, and if a hurricane hits or lightening strikes, all bets are off. Butterflies are flapping their wings all the time. Sure, what we know works 99.999% of the time, but that’s still 1.33 days a year. Luckily, there’s a lot of things with even higher reliability, but machines breaking and needing repaired, or rebooted is common. 99.99999…% wears out pretty quickly when a processor is operating at 3 GHz (3,000,000,000 operations per second), or 94,670,208,000,000,000 in a year. That’s right, 94 quadrillion times. To get that down to 2.59 failures per day you need 99.9999999999999% reliability. I think you get the point. However, when Gödel’s work is looked at with consideration of the power of shared illusions, and his proof that there will always be a true statement that can not be proved, one can see a justification for accepting the premise of circular logic in a limited capacity. Much theory is fairly secure in predicting results, but theories are just that, theories. Sure, they all break down at some point, but when the illusion fails, the products of the illusion still remain.
What we need is more efficient natural gas usage in the home, used for primarily for heating air/water, and cooking. Possibly even generating electricity. If there was a burner turning a steam turbine for electricity, the used steam could be used to heat water on the return to the boiler. The water could provide hot water for showers, and also heating in the winter… mini radiator + fan at air ducts. Of course, you have to be sure the plumbing can handle the extra flow of water needed to heat the air. Good insulation on hot water pipes would be critical.
Of course, this would work just as well with any heat source, as long as it can boil enough water. We want to maximize fuel BTUs but minimize cost and environmental impact. We can also experiment with other forms of “steam.” All liquids expand as they evaporate… For example, alcohol would evaporate at a lower temperature, requiring less heat to get things moving. However, we would have to consider the mass density of the vapor. We want to use whatever fluid -> vapor expands the most for the BTU level we will have, while also meeting any requirements for minimum temperature. For example, if all of the data centers used water cooling, they could reduce the energy usage from all those fans to one water pump per rack, and that hot water could evaporate alcohol (or something else) to run a small steam engine to turn a generator. By recouping the heat energy, you may even be able to power the pump, effectively neutralizing your cooling costs.
Also, recovering the heat from cooling devices (think freezers and ACs) would not only recover some electricity, but in the case of a freezer, could lower the load on the AC in the summer.
If it turns out that water (or something with an even higher vaporization point) is the best material, a natural gas flame could “finish” the material to the right temperature in a boiler. Anyway, it seems like it may be better than what we’re doing now. Just an interesting idea I would love to see it work.
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