Six of the most profound intellectual discoveries in human history were made in the fifty three years from 1879 to 1931.

**Table of Contents**

- 1879-1931
- The Predicate Calculus
- Different Sizes of Infinity
- Special Relativity
- General Relativity
- Quantum Mechanics
- Incompleteness
- Historial Legacy
- End Notes

**1879-1931**

The fifty three years spanning from 1879 to 1931 produced six of the most profound discoveries and intellectual insights in human history. Each of these discoveries was more than revolutionary, being extraordinary in some unprecedented way. For example, four of them were so counterintuitive that many refused to accept them (some even to this day). However, the mathematical and logical discoveries have been rigorously proven beyond doubt, and the empirical evidence for the physical discoveries is substantial. No other time period in history has produced so many earth-shattering discoveries in so short a period.

These six discoveries are as follows:

- 1879 – Gotlob Frege – Invention of the modern predicate calculus
- 1890 – George Cantor – Discovery of transfinite set theory and the proof that there are different sizes of infinity
- 1905 – Albert Einstein – Special relativity
- 1915 – Albert Einstein – General relativity
- 1905-1926 – Numerous Contributors – Quantum mechanics
- 1931 – Kurt Gödel – Incompleteness theorem.

I will describe each of these in more detail below.

**The Predicate Calculus**

Around 360 BC Aristotle invented the syllogism^{1}, a limited form of logical reasoning. Aristotle’s logic remained essentially unchanged for an astounding 2200 years, when George Boole published his laws of *Laws of Thought*^{2}, in which he outlined the foundations of what is known today as binary, propositional, or sentential logic^{3}. However, neither Aristotle’s syllogisms nor Boole’s logic was a subset of the other, and neither was even close to being powerful enough to express formally most statements in mathematics and science; some more fundamental system was required. The system that did this is what is known today as the predicate calculus, or first-order logic, and was first published (albeit with a strange notation), by Gottlob Frege, in his 1879 masterpiece *Begriffsschrift*^{4,5}.

Frege’s discover was remarkable because it was highly original, and arguably the most important contribution to logic of all time. Frege’s invention codified, in a rigorous manner, the rules of deductive thought. It was later shown that Frege’s system is both sound and complete. By “sound” is meant that any deduction that can be worked out via the system will indeed be a valid deduction. By “complete” is meant that any valid deduction can indeed be demonstrated using Frege’s system (this theorem was proven by Kurt Gödel in 1930)^{6}. In fact, much of natural language can be stated formerly using the predicate calculus and set theory (or the closely related second-order logic), falling short only with concepts of contingency, or possibility and necessity. For these the systems of Boole and Frege must be extended to modal logic^{7}.

Frege’s system is truly the foundation of modern thought. Every formal theorem derived in any rigorous subject, such as mathematics, physics, or economics, is derived using the predicate calculus. Frege’s discovery showed us how our logical thinking could be translated precisely into a formal language with rigorous rules of manipulation.

**Different Sizes of Infinity**

In 1895 and 1897, Georg Cantor published two articles on set theory that astounded the mathematical world by proving that there are different sizes of infinity^{8}. Cantor showed this by demonstrating that the set of integers, despite being infinite, cannot be matched up with all of the real numbers. In other words, no matter what method is used to match up these two sets of numbers, there will *always *be real numbers that are left out. He therefore concluded that the size of the set of all real numbers was fundamentally larger than the size of the set of all integers, despite the fact that both sets are infinite. Cantor called the number of integers , and the number of real numbers . It was later proven that = 2^{}.

Cantor’s discovery is remarkable because it is extremely counterintuitive, and most people find it difficult to accept at first. Common sense would say that infinity, if it even exists, is surely of one size only. Any infinite sequence goes on forever, and forever is, simply, forever. Many of the properties of back up this intuition. For example, + = , and indeed for any real number R, R* = . It is even true that * = , and for any integer N, ^{N} = . It is only when you raise a number to the power of that you get a larger infinity.

The logic of Cantor’s proof is undeniable, and years of work in modern set theory corroborates his result. Infinity, strange as it seems, comes in different sizes.

**Special Relativity**

The year 1905 is often called Albert Einstein’s *annus mirabilis*, or miracle year, because he published no less than four important papers in the history of physics^{9}. Their subjects included the photoelectric affect (discussed below in the section on quantum mechanics), Brownian motion, special relativity^{10}, and the equivalence of mass and energy^{11}. His paper on special relativity, arguably the most famous of the four, stated that time and space were one and the same, although the geometry of this “space-time” was remarkably different from the geometry of our everyday world. The implications of this new geometry were very strange, such as the fact that physical objects get measurably shorter, and time runs measurably slower, at speeds approaching the speed of light.

In around 300 BC Euclid proposed the first mathematical theory of geometry. This so-called Euclidean system described the common-sense geometric properties of our world. In modern times, many so-called non-Euclidean geometries have been developed^{12} that deviate from Euclid’s system in various significant ways. One of the key features of any geometry is what is called its metric, which is the formula to compute distances from a coordinate system. For example, in Euclidean geometry, the metric is the famous Pythagorean formula, which states that the distances between two points as measured in a coordinate system with axes x and y is: Δd^{2} = Δx^{2} + Δy^{2}, where Δd is the difference between the two points, Δx is the difference in their x coordinates, and Δy is the difference in their y coordinates^{13}. In Euclidean geometry, this formula holds true no matter what axes are used for x and y, as long as they are at right angles to one another, or “orthogonal” in mathematical jargon. Note that the metric also defines the curve that represents the set of points equidistant from any given point. For Euclidean geometry, this is a circle, since the formula for a circle centered (for simplicity) at the point (x,y) = (0,0), is x^{2} + y^{2} = r^{2}, where r is the radius of the circle.

What Einstein’s special theory of relativity said was that space and time are dimensions of a single space, but the metric for this space is different from the normal Euclidean one. If we use s as the “distance” in special relativity, then the formula for this distance is ds^{2} = dx^{2} – dt^{2}. For this formula^{14} to work, the distance and time units must be calibrated to the speed of light. This means that the unit of distance selected must be intuitively much large than the unit of time selected. Thus, if you select one second as your unit of time, the unit of distance must be one light-second, or 300,000 kilometers (186,000 miles)! Conversely, if your unit of distance is a meter, then your unit of time must be the time it takes light to travel one meter, or about 3 nanoseconds! This is why we don’t normally notice the affects of special relativity: they only become apparent when an object covers large distances in small amounts of time; i.e. for objects traveling near the speed of light.

What is the shape in space-time that defines a constant distance? The equivalent formula for distance from the point (0,0) is r^{2} = x^{2} – t^{2}, which is the formula for a shape called a hyperbola^{15}. A hyperbola is a curve that is round at one end (near the point (0,0)), but open at the other end. As the special relativity hyperbola heads out from (0,0), it approaches, but never touches the two lines that represent the speed of light in both directions in the x dimension of space. This implies four remarkable things:

- Nothing can go faster than the speed of light.
- The speed of light is the same, no matter how fast you are going.
- As objects approach the speed of light, strange things will seem to happen to them (like their getting shorter and time slowing down for them).
- “Now” is not universal. If two events at two different places seem to one observer to take place simultaneously, for a second observer traveling at a finite velocity relative to the first, one of these things will seem to take place earlier than the other
^{16}.

Special relativity was remarkable because, like Cantor’s discovery, it ran counter to our intuitive understanding of the world. Before Einstein, time was independent of space, and completely absolute^{17}. For everyone everywhere, time was the same, and marched inevitably forward at the same rate. Einstein’s theory proved this intuition to be incorrect. Time is the same as space, and the two can be interchanged: accelerating to a different velocity is like a “rotation” in space-time. Time does not progress at the same rate everywhere. And perhaps most difficult to accept, “now” is not the same for everyone: what is happening now for me at a distant location is not the same as what is happening now for someone traveling at a finite velocity relative to me.

**General Relativity**

When Einstein made his presentation on general relativity to the Prussian Academy of Science in 1915^{18}, he astounded the world again, this time because his new theory was completely unexpected and unnecessary. As Thomas Kuhn famously documented in his classic *The Structure of Scientific Revolutions*^{19}, revolutions in scientific thinking are normally driven by necessity. Experimental data tends to accumulate that challenges the prevailing theory. This theory tends to be modified to accommodate the new data, but as the modifications accumulate the theory gets messier and messier. Eventually a new theory is proposed that better explains the data and replaces the old one.

Such was not the case with the predecessor to general relativity, Newton’s law of universal gravitation. Newton’s law pretty much explained everything about gravity that had been observed, with the exception of a few minor anomalies such as an observed shift in the perihelion of the orbit of Mercury. Most astronomers at the time thought that these anomalies could be explained by the presence of as-yet-unobserved astronomical bodies. It is not clear that Einstein was even aware of these problems. What *did *bother Einstein was that Newton’s law required the affects of gravity to travel instantaneously, and in particular, faster than the speed of light, which his special theory of relativity said was impossible^{20}.

The theory that Einstein developed was shocking in its difficulty, beauty, and scope. Not only was it consistent with special relativity, and correctly described the reason for the shift in the perihelion of the orbit of Mercury, it also made a remarkable number of predictions that astounded both scientists and laymen alike, among these being:

- Light is affected by and can be bent by gravity.
- Time travels slower in gravitational fields than in the absence of them.
- Any mass concentrated in a small enough volume can lead to a “singularity”, or region of infinite density known as a black hole.

Copious experimental evidence exists for each of these predictions. In addition, general relativity is the foundation of modern cosmology, and in particular was a prerequisite for the prevailing theory of the “big bang”, and its subsequent period of “inflation”^{21}.

Despite it’s success, most cosmologist think that general relativity will eventually be replaced^{22}, because right now there is no known way to reconcile it with quantum mechanics. Nonetheless, general relativity is remarkable for two reasons. The first was the manner in which is was devised. There may not be any other scientific theory that was presented with so little experimental need to replace its predecessor. The second reason is the scope of its predictions and their affects on science. Our entire understanding of modern cosmology, including the big bang, the expanding universe, and black holes, is based on Einstein’s second amazing theory.

**Quantum Mechanics**

One of the other papers that Einstein published in 1905 described his theory of the photoelectric affect, which explained why light hitting certain surfaces caused electrons to be emitted from the surface with constant energies^{23}. Einstein sought to explain this by proposing that light exists only in discrete bundles of energy that he called “quanta”. Einstein’s discovery was the first in a series of discoveries by numerous physicist^{24} which eventually came to embody what we know today as quantum mechanics. Einstein won the Nobel Prize in 1921 for his contributions to physics, with his theory of the photoelectric affect being the only one specifically mentioned (there was no mention of relativity at the time because it was still very controversial)^{25}.

Quantum mechanics is another discovery from this time period that is remarkable because it is so counterintuitive. It is the most accurate theory every invented^{26}, and yet is widely criticized as making no intuitive sense. There are three facets of quantum mechanics that seem most difficult to accept:

- Indeterminism – Quantum mechanics predicts that the world is not deterministic, but that certain things happen completely at random. This randomness is not a function of our inability to understand the details of the process, as is the case with predicting the weather or the stock market, but is fundamental to the physics of the universe itself.
- Nonlocality – As mentioned above, Einstein’s theory of special relativity states that nothing can go faster than the speed of light. However, quantum mechanics seems to violate this prediction of special relativity by the fact that under certain circumstances particles seem to be able to communicate their states to one another instantaneously. While strictly speaking this is not a violation of special relativity, since this phenomenon cannot be used to send information faster than the speed of light, it is strange nonetheless.
- The role of consciousness in measurement – The so-called Copenhagen interpretation
^{27}of quantum mechanics calls for a special role of an observer, and often this is thought of implying that human consciousness has a special role in quantum mechanics.

In my essay on Quantum Theory, I have argued that the first two of these are necessarily true, and that the third is wrong and therefore will be modified as time goes on. Note that this will not invalidate quantum mechanics as a theory, as this third item is a result of the prevailing *interpretation *of the theory. Despite this, it cannot be denied that quantum mechanics is one of the strangest, most counterintuitive theories every invented, and for this reason is one of the most remarkable theories in scientific history.

**Incompleteness**

Since the time of Euclid described above in the section on special relativity, mathematicians have always assumed that any branch of mathematics could be reduced to a small set of postulates or axioms, and that the theory based on this set of axioms would be complete, meaning that every relevant statement that was true could be proven by the axioms. In 1900, David Hilbert issued a series of challenges to the mathematics community, including his second challenge that asked to prove that the axioms of arithmetic are consistent^{28}. The prevailing thought at the time was that mathematics itself was both complete and consistent, and it was just a matter of time before this was proven to be true.

In 1931, Kurt Gödel shattered these hopes by proving that this was impossible via his famous incompleteness theorem^{29}. In particular, what he demonstrated was that any sufficiently complex axiomatic system (including arithmetic and set theory), if consistent, could not possibly be complete: there would always be statements that could not be proven from any finite set of axioms. In short, he showed that truth is broader than proof; not all truths can be proven. Very roughly, he did this by constructing a mapping from statements about arithmetic and logic to arithmetical statements themselves, and created an arithmetic statement that, when thus translated, said in essence: “I cannot be proven”^{30}.

Gödel’s proof was remarkable for the ingenuity that he used in the proof, the fact that it was unexpected, and the fact that it was another example of a fundamentally counterintuitive discovery. Since the time of Euclid, mathematics was thought of as perfect and neat — every truth could be proven from a small set of axioms, with no contradictions. Furthermore, it was thought to be just a matter of time before this ultimate set of axioms had been discovered. Despite the fact that naive set theory had been found to be inconsistent^{31}, there were still high hopes that all of mathematics could be embedded in some complete form of set theory. Gödel’s proof forever destroyed this dream.

**Historical Legacy**

It is hard to believe that so many closely held, intuitive ideas and conceptions about mathematics, logic, and the universe could be overturned in such a short period of time. Yet, this did indeed happen. In 1932 the world was, intellectually, a much different place than it was was in 1878. I doubt that history will ever see another period of time in which so much of human thought changes so quickly.

- “Sylogism”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/Sylogism>. - “The Laws of Thought”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/The_Laws_of_Thought>. - For a good introduction to sentential logic, and logic in general, see Enderton, Herbert, B.,
*A Mathematical Introduction to Logic*, Academic Press, San Diego, CA, 1972. - “Begriffsschrift”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/Begriffsschrift>. - This history of logic is pretty much taken verbatim from Glymour, Clark,
*Thinking Things Through*, MIT Press, Cambridge, MA, 1998. - Enderton,
*op. cit*., pgs. 124 & 128. - “Modal logic”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/Modal_logic>. - See Cantor, Georg,
*Contributions to the Founding of the Theory of Transfinite Numbers*, Dover Publications, New York. - “Albert Einstein”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/Albert_Einstein>. - Einstein, E., “On the Electrodynamics of Moving Bodies”, Annalen der Physik, Vol XVII (1905), pg. 891-921. For an English translation available online see: http://www.fourmilab.ch/etexts/einstein/specrel/www/.
- This is Einstein’s famous formula E = mc
^{2}, which was a consequence of the theory of special relativity. - “Non-Euclidean geometry”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/Non-Euclidean_geometry>. - The same formula is used to calculate the length of the hypotenuse of a right triangle, and the two sides of the triangle can be the basis for a coordinate system. The coordinate axes must be at right angles, or “orthogonal”.
- I have included only one normal space dimension x here, but this formula can be extended to all 3 spacial dimensions as ds
^{2}= dx^{2}+ dy^{2}+ dz^{2}– dt^{2}. - For more information on hyperbolas, see the Wikipedia article: “Hyperbola”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/Hyperbola>. - This phenomenon is known as “relativity of simultaneity”. For more information see “Relativity of simultaneity”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/Relativity_of_simultaneity>. - For more on this see my essay on Time.
- “General relativity”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/General_relativity>. - Kuhn, Thomas, “The Structure of Scientific Revolutions”,
*International Encyclopedia of Unified Science*, Vol. 2., No. 2, University of Chicago Press, Chicago, IL, 1962. - Wudka, Jose, “The happiest thought of my life.”, University of California, Riverside, September 24, 1998, URL=<http://physics.ucr.edu/~wudka/Physics7/Notes_www/node85.html>.
- “Inflationary cosmology”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/Inflationary_cosmology>. - This is not true of special relativity, which I have heard it called by at least one physicist (whose name escapes me) a “super theory”, in that it may never be challenged. The second law of thermodynamics may be another super theory.
- “Photoelectric effect”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/Photoelectric_effect>. - For details see the Wikipedia article on the history of quantum mechanics: “History of quantum mechanics”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/History_of_quantum_mechanics>. - “The Nobel Prize in Physics 1921”,
*The Nobel Foundation*, URL=<http://www.nobelprize.org/nobel_prizes/physics/laureates/1921/>. - Orzel, Chad, “The Most Precisely Tested Theory in the History of Science”, ScienceBlogs, May 5, 2011, URL=<http://scienceblogs.com/principles/2011/05/05/the-most-precisely-tested-theo/>.
- “Copenhagen interpretation”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/Copenhagen_interpretation>. - “Hilbert’s problems”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/Hilbert%27s_problems>. - For a summary and references see the Wikipedia article on the theorem: “Gödel’s completeness theorem”,
*Wikipedia*, URL=<http://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem>. - For an excellent layman’s description of Gödel’s proof, see Penrose, Roger, the
*Emperor’s New Mind*, Oxford University Press, New York, 1989, pg. 105-108. - For a further discussion of various set theories, see my essay on metaphysical Universals.

November, 2013