On Websites at Florida

I have moved my faculty website to this website. My personal and faculty website are now located and combined here at saemiller.com. There is a redirect from https://faculty.eng.ufl.edu/fluids/

The university depends on academic freedom, and academic freedom depends on tenure. Without tenure there is no academic freedom, and without academic freedom there is no university. 

Gödel and Time

$\mathrm{LL}$ cosmological solutions with non-vanishing density of matter known at present ${ }^1$ have the common property that, in a certain sense, they contain an “absolute” time coordinate, ${ }^2$ owing to the fact that there exists a one-parametric system of three-spaces everywhere orthogonal on the world lines of matter. It is easily seen that the non-existence of such a system of three-spaces is equivalent with a rotation of matter relative to the compass of inertia. In this paper I am proposing a solution (with a cosmological term $\neq 0$ ) which exhibits such a rotation.

Kurt Gödel, Institute for Advanced Study, Princeton, New Jersey

Gödel circumvented the light-speed barrier by suggesting that a fast-spinning object could distort space and time, making their properties coalesce. At sufficiently high spins, returning to the starting point in space would coincide with returning in time. The limitation of Gödel’s concept is its reliance on a spinning universe, a condition not supported by current evidence. But, who knows if such a place exists.

Reference

Gödel, K. (1949). An example of a new type of cosmological solutions of Einstein’s field equations of gravitation. Reviews of modern physics, 21(3), 447.

Underlying Assumptions

How Can Any Part of Mathematics Be Proven? The answer lies in demonstrating that a mathematical statement must be true if the underlying simpler math is also true. It is a great difficulty to show the increment from 1 to 2. Between 1910 and 1913, a three-volume work was published on this subject. Titled Principia Mathematica (The Principles of Mathematics), it imitated Isaac Newton’s 17th-century research. Its aim was to establish the fundamental basis of mathematics through logic. Authored by celebrated British philosophers Bertrand Russell and Alfred North Whitehead, this work is a cornerstone of the philosophy of mathematics. The first volume outlines the approach for the subsequent volumes, focusing on logical type theory. In type theory, every mathematical object is categorized within a hierarchy of types, each a subset of those above it. This categorization aims to prevent paradoxes, which often arise in logical systems. The second volume examines numbers. The third volume covers series and measurement. Despite its excellence, Gödel’s theorem would soon reveal that any attempt to prove the entire system of mathematics logically, including Russell and Whitehead’s effort, is itself a logical impossibility.

References

  • Russell, B., & Whitehead, A. N. (1910-1913). Principia Mathematica. Cambridge University Press.
  • Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica.
  • Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38(1), 173-198.

First Website

I recently took a trip down memory lane by looking at old websites. I had forgotten that my first website was published in 1996. It appeared on the University of Michigan EECS server for artificial intelligence. I was only in high school but had early JavaScript for a random quote generator. That was version 5. Later versions of the website had built-in Virtual Reality Markup Language (VRML), allowing users to spin a three-dimensional cube in outer space and click different sides to access various sub-websites. It was really cool for the time—I haven’t seen anything quite like it even today.

Unfortunately, the hypercube and VRML are lost to time. I’ve tried to find them, but I think there’s nothing in the Internet Archive, nor have I found anything on old hard drives. I did have a major hard drive crash in the early 2000s without a backup. That was probably the only copy. But I still remember creating it in high school on an old Pentium 90 computer, or perhaps something earlier.

What I really reflect on is how different the internet was in the 1990s compared to today. Today, the internet is a corporate landscape, but back then it was like the wild west of freedom and information theory. My website even had a blue ribbon for the free and open information campaign of the internet. I think it was the Blue Ribbon Campaign, but I’m not quite sure. I also recall that I was very interested in online gaming, and some of my early thoughts and memories about gaming were on the website. There were a few other websites I made that were lost to time, like everything else, in an unfortunate digital loss.

Boole and Laws of Thought

George Boole, in the 1840s, proposed that variables could represent more than just numbers. Boole’s work, published in “An Investigation of the Laws of Thought” (1854), introduced algebra with two values: 1 (true) and 0 (false). Instead of traditional algebraic operations, Boolean algebra uses AND, OR, and NOT, also known as conjunction, disjunction, and complement. Conjunction (∧) is like multiplication, with any 0 resulting 0 (false). Disjunction (∨) is similar to addition, but 1∨1 is defined as 1. Complement (¬) exchanges values, swapping 0 for 1, and vice versa. These operations can be expressed in various ways, including truth tables and Venn diagrams, which show their relation to sets of x and y (varying groups of 1s and 0s). Boole derived other operations from composites of these basic ones. In the 1930s, Claude Shannon used Boolean equations to control switching circuits, creating the first logic gates, in the form of thermionic diodes. A logic gate can use anything as an input.

References:

Boole, G., 1854. An investigation of the laws of thought: on which are founded the mathematical theories of logic and probabilities (Vol. 2). Walton and Maberly.

Adhémar Barré de Saint-Venant on Flow

If the velocities [of water in rivers] remained constant in each point of the traversed space, the surface of the liquid would look like a plate of ice and the herbs growing at the bottom would be equally motionless. Far from that, the stream presents incessant agitation and tumultuous, disordered movements, so that the velocities change in an abrupt and most diverse manner from one point to another and from one instant to the next. As noted by Leonardo da Vinci, Venturi, and especially Poncelet, one can perceive eddies, large and small, with a vertical mobile axis. One can also see, at the surface, bouillons, or eddies with a nearly horizontal axis, that constantly surge from the bottom and thus form genuine ruptures, with the intertwining and mixing motions that M. Boileau observed in his experiments.

Adhémar Barré de Saint-Venant, 1872.

Fechner-Weber

The Fechner-Weber law states that for a sensation’s intensity to increase in an arithmetic progression, the stimulus must increase in a geometric progression. This relation describes sensory perceptions and physical stimuli for hearing. Human hearing can detect noise so quiet that the eardrum moves less than an atom’s width, and noise 10 trillion times more powerful. Similarly, the faintest star we can see is about 10 trillion times less powerful than the Sun. In a noisy environment, we can distinguish quieter sounds, such as conversation or a dropped coin. Senses respond not to the absolute increase in a stimulus, but to fractional increase. Ernst Heinrich Weber showed in 1846 that the change in a person’s perception of weight was proportional to the logarithm of any increase. For acoustics, as a stimulus becomes ten times stronger, the perceived increase only doubles. Gustav Fechner elaborated on Weber’s discovery in 1860. The Fechner-Weber law can be experienced by halfing noise. If our sense of hearing was linear with respect to intensity, that would reduce the noise by half. However, although the sound power in the room has halved, the difference in the volume we hear is barely noticeable.

References:

Weber, E.H., 1851. Annotationes anatomicae et physiologicae: Tractatus de motu iridis summa doctrinae de motu iridis. Koehler.

Fechner, G. T. (1860). Elemente der Psychophysik.