Mathematics and science | Douglas Wilhelm Hard

Mathematics is an abstraction that allows us to model reality, or at least, that is how it started. The first abstraction was integers: a means of counting objects for the purpose of trade, business interactions, records and taxes. Another early branch of mathematics was geometry formalized by Euclid et al., and then Aristotle et al. introduced and formalized logic. After integers came ratios, or what we would consider today to be rational numbers. The ancient Greeks even showed that the square root of two was not rational, so there were measurable distances that could not be expressed as a ratio. The area of a unit circle is π, so one question was given a circle, can you create a square of the same area (can you square a circle) using a compass and a straight edge. Essentially, this asks if is π rational or the root of a polynomial with integer coefficients (the roots of a polynomial with rational coefficients are the same as the roots of that same polynomial multiplied by the least-common multiplier of the denominators of the coefficients, so it is not necessary to consider polynomials with rational coefficients, one need only focus on polynomials with integer coefficients).

Calculus was invented to help model the physical universe, with Newton and Leibnitz.

In the 1800s, the axiomatic method became more universally applied to more than just geometry, and soon all of mathematics became axiomatized. First and foremost, logic was axiomatized, with the introduction of Boolean logic and algebra. Here, there were only two values, true and false,

Statements in mathematics can be proven: given a set of axioms, together with Boolean logic and set theory, one can then proceed to apply the axioms to prove additional statements, and if a statement is proven using the axioms, that statement is called a theorem. It is fascinating in Boolean logic and algebra that you only have a small set of axioms that describe the basis of Boolean logic and algebra, and then all other properties, or theorems, can be deduced from this small set of axioms using those axioms. Specific statements in Boolean logic and algebra are called tautologies.

Similarly, set theory is derived from a small set of axioms and from those axioms together with Boolean logic and algebra, one can deduce many true statements or theorems about sets.

There are concepts about equality that also go into the past: Aristotle realized that x = x is a fundamental axiom of equality. Euclid deduced that the transitive property of equality must also be true: if x = y and y = z, then x = z. It has since been deduced that the properties of equality can all be deduced from four axioms:

Equality is reflexive: x = x for all objects under discussion.
Equality is symmetric: x = y if and only if y = x.
Equality is transitive: if x = y and y = z, then x = z.
Equal objects substituted into true statements continue to leave those statements true.

An example of the latter is any true statement concerning the polynomial x² - 1 must also be true if we substitute (x - 1)(x + 1).

Initially, systems such as the natural numbers (0, 1, 2, 3, ...) were described using a different set of axioms: the Peano axioms that posit that given a natural number n, it is always possible to find a successor to that natural number, and we will designate that success by S(n). Also, equality for the natural numbers satisfies the axioms of equality specified above. For example, S(5) = 6. With this operation, we then have the following four axioms:

0 is a natural number.
For every natural number n, S(n) is a natural number. That is, the natural numbers are "closed" under the successor operation.
For all natural numbers m and n, if S(m) = S(n), then m = n. That is, if n is the successor of m, then m is unique: n cannot be the successor of two different natural numbers. This says that the successor operation is one-to-one.
For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.

In mathematics, the collection of a set of axioms together with all theorems that have to this point been deduced from those axioms is described as a theory. For example, number theory includes the axioms that describe the integers together with all theorems that may be deduced from those axioms. A statement that has not been proven to be true or false (that is, a statement that has not yet been elevated to the level of a theorem while also not having been shown to be false) is called a conjecture (sometimes also called an open question). There may be conjectures within a theory; for example, for a long time, it was an open question as to whether or not Fermat's Last Conjecture was indeed a theorem (to call it a theorem before it was proved is a misnomer). Once Andrew Wiles proved it, it could then honestly be called Fermat's Last Theorem.

Now, Russell and Whitehead demonstrated convincingly that arithmetic can be based only on logic and set theory. What this says is that we actually do not need Peano's axioms; however, to deduce everything from the axioms of logic and set theory would be pointlessly difficult and arduous. We continue to use our higher axioms as they form a reasonable basis for future deductions.

Beginning in the 1800s, mathematicians came up with axioms that did not seem to be based in reality or took subsets of previous sets of axioms. There were projective geometries, and non-Euclidean geometries, and affine geometries. At some point, it was thought that all of mathematics could be deduced from given statements; however, this was shown to be false by Gödel. I'll try to explain this next.

Theorems are all that we can prove. We assume some axioms are true, and then we use logic to demonstrate the truth of other statements from those assumed axioms. Mathematics may be used to model reality, but it does not prove reality. We don't necessarily require proofs in mathematics; after all, we could just have observations: (x + y)² = x² + 2xy + y² is true whenever x and y are integers, rational numbers, real numbers and complex numbers, but not when x and y are square matrices. If all mathematics is is making observations, this would be disappointing, though. Thus, instead, we postulate a much smaller set of axioms and then prove other results based on that smaller set of axioms. It would be great if all true statements about reality can be proved simply by determining all consequences of a small set of axioms, but that is simply not the case: Gödel proved that given a set of axioms from which one can deduce the ability to count, then there are true statements that cannot be proved from those axioms. If you then assume a statement is true because you cannot prove it, at some point, you must end up with contradictory results no matter how hard you try.

For example, we know that we can prove that there are infinitely many prime numbers. A composite number is any number that can by divided by more integer between 1 and itself. A prime number is any number that is not composite. However, no one has proved that there are infinitely many prime pairs; that is, infinitely many prime numbers that differ by two; for example, (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), ..., (7877, 7879), etc. are all prime pairs. Now, suppose we determine that we cannot prove that there are infinitely many prime pairs. Ali may then add the additional axiom: there are infinitely many prime pairs, while Bailey may add the additional axiom: there are only a finite number of prime pairs. Assuming either of these cannot lead to a contradiction, because if either did, then that would prove the other as a result of our current axiomatic system.

Mathematicians did this with the axiom of choice: you could assume that the axiom of choice is true, or you can assume it is false, and the mathematics that you deduce from either assumption leads you to two very different mathematical models. Assuming it is true (adopting it to be an axiom) produces results such as it being possible to split a sphere into 41 pieces, moved and rotated without ever overlapping, and then reassembling these pieces into two spheres of the same volume as the original. For example, consider the Tangram puzzle. It is possible to move and rotate these pieces of a square on the plane so as to produce two squares, each of half the area of the original. You cannot, of course, ever accomplish this with a classic interlocking puzzle because you cannot separate two pieces without lifting one into the third dimension. Trevor Wilson, however, showed you can split a sphere into 41 pieces, move and rotate them without ever blocking each other, in such a way to produce two spheres that are equal in volume to the original, but for each point in each of the two spheres, there is a point in the original sphere from which it came. Personally, I prefer to assume that the axiom of choice is false. After all, we cannot create a sun out of a pea by splitting the pea into pieces, reassembling them, and then repeating the process with each resulting pea at most 114 times. Consequently, any theorem that is proved using the axiom of choice cannot provide a model for reality: we can only split peas.

Another case may be the question of whether or not P = NP; that is, can every Boolean-valued problem (a problem to which there is a yes-no answer) were a solution can be verified in polynomial time, be also solved in polynomial time. For example, given n cities, is it possible to visit all n cities by travelling less than 1203 km? Given a list of n cities, then you can check whether or not this is less than 1203 km just by adding together. Currently, however, there is no algorithm that will give you that list of n cities assuming a solution exists in less than 2ⁿ steps, at least, in the worst-case scenario. I personally believe that P ≠ NP, but this is simply the consensus view of the majority of computer scientists, and I certainly have nothing useful to say about this problem.

Mathematics is a field that is generally deductive:

We cannot prove anything about reality; at the very best, we can make a statement that describes reality and try to show that that statement is false. This is the scientific method.

The scientific method

We look at the world and we observe patterns: things fall down, sky is blue, the world looks flat, rain comes from the sky, people who have multiple sexual partners tend to die from horrible diseases, and the winds in Southern Ontario seem to predominately come from the northwest. You can now make a guess as to why any of these are the case.

Aristotle hypothesized that everything in the world is made of one or more of four fundamental elements: earth, water, air and fire. Each of these has a natural order (earth below water, which is below air, which is below fire) and these elements have a tendency to move back to their natural order.
Some ancient philosophers and authors noted that the sky is the same color as tarnished bronze, and thus hypothesized that the sky was a bronze canopy.
Many ancient authors, including likely all authors of the Tanakh, believed and some very stupid or credulous (or both) people today believe that the world is flat.
Many ancient authors, also including likely all authors of the Tanakh and at least one author of a letter in the New Testament, believed that there was water above that (bronze?) dome in the sky, and this water falls to Earth through holes in that dome.
Until recently, most people believed sexually transmitted diseases were a curse from the gods.
At least one indigenous leader from Southern Ontario today proclaimed that the northwest wind was the dominant wind throughout the world.

Some ancient authors, however, believed the Earth was a sphere and not flat.

Thus, if we make an observation about the real world, and then hypothesize as to why that is the case, how can we determine if that hypothesis is true? Previously, perhaps you believed some explanation of a natural phenomenon is true because someone said so or because it was written down somewhere (the argument from authority). In other cases, explanations were believed because everyone else believed the explanation (the argument from popularity). However, even in ancient times, examples were seen that attempted to break away from such traditions and to instead use a more rational approach to modelling the world:

The Edwin Smith papyrus dated to around 1600 BCE and includes a description of the process of of examining the patient, diagnosing the disease, treating that disease, and the subsequent progression of the disease as a result of that treatment to the determine cures.
The philosophers of the neo-Babylonian empire came up with mathematical models that described astronomy.

Having such tools began to reveal certain truths about the universe, but these tools were not applied universally and did not necessarily prevent false hypotheses from being believed to be true. After all, it took almost two millennia before the heliocentric model became accepted, for prior to this, the common belief was that planets moved according to the motion of epicycles (planets spinning around a point where that point is orbiting the Earth) and it would take even longer before it was finally determined that leaches and arsenic do not cure ailments.

The scientific method requires a number of steps

Science

For example, Newton used calculus to model the the effects of gravity in the physical universe, and from this he found equations that appeared to approximate the effects of gravity as was actually observed in the physical universe. It was only after close observation with more precise instruments that, for example, the properties of the orbit of Mercury, it was determined that Newton's model of gravity was an imperfect model of reality: gravity seemed to be doing something other than just obeying the inverse square law. It was from this observation that Einstein proposed his theory of general relativity as a better model of the physical universe, at least, with respect to gravity. Einstein's general relativity explained the abrogation that was seen in the orbit of Mercury, but that was insufficient to show general relativity was correct. Instead, Dyson and Eddington journeyed to South America to take photographs of a total solar eclipse. Newton's theory predicted a specific deflection of the light as it passes around the Sun, and Einstein's theory doubled that deflection; the experiment supported Einstein's theory. Today, over and over again, subsequent observations and experiments demonstrate that Einstein was correct. Unfortunately, the newspapers often say that "Einstein was proved correct," but this is false: general relativity is a model of gravity. It does not say why gravity works that way, nor is it justified. Instead, it is a mathematical model that appears, given all the observations and experiments that have been run to date, to describe gravity.

Prior to more precise measurements of the orbit of Mercury, Newton's theory of gravity appeared to be a valid model of gravity; and in the next ten years, or perhaps the next century, an experiment or observations may run contrary to what is predicted by Einstein's model. This would, have two immediate consequences: first, scientists will try to confirm that the experiment or observations and try to determine additional experiments or observations that may determine exactly how Einstein's model does not match actual observations, and second, scientists will try to come up with a new model that matches both all known observations together with matching the new observations. But once again, just coming up with a newer model will not guarantee correctness. Instead, scientists will then look at the model, and try to find another previously unknown observation where the theory of general relativity suggests we should see x, and this new theory suggest we should see y (with x ≠ y), and then that experiment will be run (likely at great expense) and we will see if that new theory actually holds up!

Interestingly enough, of course, is that Newton's theory is generally good enough, and only begins to fail in the vicinity of very massive objects.

In mathematics, a theory is a set of axioms and all theorems and conjectures associated with those axioms. In science, a theory is a model of reality together with all consequences and predictions of that model.

Quantum mechanics is another theory that models the very small with, interestingly enough, complex-valued functions of one or more real variables, those real variables representing space and time. These complex-valued functions must obey certain properties proposed by the theory of quantum mechanics. However, the real world is not complex-valued functions, but complex-valued functions appear to model reality very well, at least at the atomic level. We cannot "prove" that quantum mechanics is correct, but we can try as hard as possible to show that a prediction it makes is wrong, in which case, someone will have to come up with a newer theory.

What is fascinating is that this method of observation and modelling has been used throughout antiquity: the ancient Greeks observed that:

The stars you saw on any day differed depending on where you were, and this required detailed observations made at many different locations and then someone had to collate and compare these observations.
On the Summer Solstice, the light of the Sun shown straight down a well in Syene but in Alexandria, this never occurred.
The hull of a ship disappeared under the horizon long before the mast itself disappeared, too, all at distances where that ship should still have been visible. Additionally, at ports in the vicinity of cliffs, the ships could be seen from the cliffs much longer than they could be seen from sea level.
It was determined that lunar eclipses were the result of the Earth passing between the Sun and the Moon, and the shadow of the Earth was round.

The best model that the ancient Greeks could come up with was that the Earth was a sphere. They did not understand why this was the case, but the model worked. It was not until Newton's theory of gravity that an explanation was given for why the Earth was a sphere. Later still, more accurate measurements showed that the Earth is not a perfect sphere (much to the chagrin of ancient Greeks), but rather an oblate spheroid: the rotation of the Earth causes the Earth to bulge at the Equator. If the Earth was the size of a soccer ball, the bulge at the equator would be approximately 0.75 mm. Note that an oblate spheroid is a well-defined mathematical object that not only has different diameters between the "poles" and along the "equator", but that the shape is well defined for all "latitudes." There are other descriptions of "flattened spheres," but the mathematically defined oblate spheroid is that description that best matches what we observe in reality.

A proposed model of reality that does not yet have supporting evidence is called a "hypothesis" (similar to a conjecture for a mathematical theory). It is only when a hypothesis makes sufficient predictions about reality that the model will be an accepted model, and if a model describes a sufficiently large body of observations, that model is described as a scientific theory.

To jest, one hypothesis to describe the fossils that are found in nature is that the pressures of the Great Flood recorded in the Torah that killed almost all animals, buried them in the sediments that were first dislodged by the vast amounts of waters as "all the fountains of the great deep burst forth, and the windows of the heavens were opened" and then the subsequent pressures of eight kilometers of water created both the sedimentary rock and the embedded fossils. This hypothesis even predicts that some animals such as the big dinosaurs would die first, while other more nimble animals such as mammals would survive longer, and therefore be higher up in the sedimentary rock (all fossils are formed in sedimentary rock, and few fossils survive the heat and pressure that convert sedimentary rock to metamorphic rock). Unfortunately, not all dinosaurs would die first, and all mammals die last: surely at least some mammals died very quickly during the flood, and some dinosaurs (especially those that could swim) could survive much longer than those non-swimming mammals that would drown much more quickly in the rising waters (together with all the pregnant women, newborns, infants, toddlers, children, etc.). Under this hypothesis, there should be a distribution of fossils throughout the sedimentary layers, and yet, the evidence is that only very specific layers contain certain fossils. On the other hand, a very simple refutation of the Theory of Evolution would be "fossil rabbits in the Precambrian"; that is, fossilized rabbits in a very well-defined sedimentary layer. Under the Great Flood hypothesis, we should find at least some mammals in the lower layers, but in all this time, none have been found. Additionally, the Great Flood hypothesis does nothing to explain the numerous fine-grained volcanic ash beds that contain well-preserved and finely detailed fossils, all with many feet of sedimentary rock in between, with distinct fossils within each ash bed.

Notice that in science, there are models, and those models attempt to explain what has been observed. We cannot "prove" that a model is correct, but what we can do is try as hard as possible to prove that a specific model is wrong. The Continental Drift Theory has since been shown to be based on an incorrect model, and has since been superseded by the Theory of Plate Tectonics. This does not mean that those who proposed the hypothesis of continental drift were stupid: they looked at the observations, and proposed a hypothesis that matched those observations, and to a limited extent, that model did make predictions that were correct; however, as time went on, more observations did not match the predictions of that hypothesis, and so that hypothesis was discarded to be replaced by a better hypothesis based on the movements and interactions of massive oceanic and continental (collectively tectonic) plates.