Could humans do mathematics if they were blind, deaf and senseless?

PRAXEOLOGY AND MATHEMATICS

Mathematical, logical, physical and praxeological laws are abstract rules which define the structure, boundaries and behaviour of our world.

These rules are discovered and created by humans. It seems counter intuitive to think that something can be discovered and created at the same time. Distinction between discovery and creation seems easy to grasp. Commodities like copper are DISCOVERED by explorers in the Earth crust. They can further be used to CREATE another physical objects like a copper wire. In this example discovery and creation are clearly two distinct actions.

Let’s go back to the origins of mathematics and try to think how FIRST mathematicians discovered and created mathematical laws. There was no mathematics textbooks, no axioms, no theorems. There was no teachers who could teach the first mathematicians. For that reason, he must have somehow discovered it himself.

geometrician

Fundamental question is: could he ever discover/create any mathematical laws if he didn’t have access to the physical reality through senses: sight, touch, smell, hearing and taste.

I think he couldn’t. If it wasn’t for one of his senses he would never realize even the most basic algebraic notions like existence or non-existence of an object. He wouldn’t have realized that there could be one item, two items or several items of something. If Euclides couldn’t see, he would never realize that “a straight line segment can be drawn by joining any two points”. Only because he could observe (and further abstract the notion of a point, or a line segment) he was able to DISCOVER the first axiom of geometry. He discovered it using senses and abstraction, but he CREATED the axiom by verbalizing it in the human understandable language and notions. This is how mathematicians are both discoverers and creators of mathematical laws at the same time. They discover them either through observation of physical reality or through deduction. However, this second method can only be used if initial DISCOVERY and CREATION took place beforehand.

To make the distinction between discovery and creation even more distinct, think about the world where the physical objects are exactly the same as in our world, but humans have no senses, therefore no means to observe these physical objects. Even though a mathematician can discover and create mathematical theorems without observing reality, by deducting them from already established axioms and theorems, he could never started this process if he didn’t derive his first abstractions from observation of the physical reality. 1 + 1 equalled 2 before the first human realized it does equal 2, and also before the first human ever appeared in our universe. This simple realization is the foundation of the mathematical realism (or metaphysical realism in the broader sense).

Would the physical objects and the mathematical laws which “govern” them still exist? Yes, they would exist, but humans would not be able to discover them and would not be able to create a human understandable abstraction of these laws.

These laws therefore exists in two version. Firstly, they exist, operate and act as the code of the physical world. In the second version, they exist as human understandable approximation of the first version.

There are at least two ways for the geometrician to assume confidence in his way of thinking and doing (deriving) further geometrical theorems.

Firstly, a mathematician can compare the results of his own independent work (work of deriving new theorems), with that of another mathematician. If he did this for the first time, he would be astonished that at least some theorems (which he derived himself through deduction) had also been derived by other mathematicians. This is because, the mathematical laws do not exist in his own mind, or the minds of the other mathematicians, but because they exist as the laws of the world they both happen to be living in. Let’s take an example of two geometricians discovering and creating geometrical theorems. They both started discovering the theorems from the same starting point – five axioms of geometry, using the same notions or terminology and the same method. The interesting, intellectual “by-product” of this method is that the discovery of the laws does not require to “consult” the experience of the physical world. Moreover, the physical world would never provide an exact, physical example of a straight line, a point or any other geometrical figure. And still all mathematicians know that the underlying mathematical laws are true and they constitute the fundamental laws on which the physical world is constructed. For those who are accustomed to think in the language and realm of computer object-oriented programming, the class definition would resemble the mathematical theorem, while the actual physical world would be an instance of that class. This of course leads us to realize that another instance of that class could possibly exist. This other instance would be an example of “another world”. The world could be different in regards to the variables values, objects existing in this world, but it would similar be in regards to structure and boundaries defined in the definition of this class (class code).

Second way, a mathematician can assume confidence in his work is through practical application of mathematics – engineering. The sole fact that engineers use mathematics to build constructions like building, bridges or roads can give us a pretty strong (but not absolute) confidence that the axioms and theorems used in mathematics (employed to calculate physical characteristics of the construction during design phase) are indeed true and they approximate the mathematical rules of the world well. Those who have used building, roads, railways, computers, satellites, cars, planes, networks, etc and still hold that “truth doesn’t exist, it’s all relative”, or “truth is relative to the used language and does not reflect the reality”, plus several other relativist and postmodernists claims should take the train ride on a bridge over a canyon, a bridge not build on axioms and theorems of mathematics but “Designed based on the postmodernist, fluid ecosocial discourse which includes feelings and needs of the minorities previously underprivileged” (thanks Ayn Rand).

bridge

PRAXEOLOGY AND PHYSICS

Let’s think about a physicist who sets up an experiment. His goal is to prove a gravity law. He sets up a lab, he takes an iron object, he measures the weight of the object, he knows the gravity force on earth, and then he calculates the time it takes for the object to move when the object is dropped from 2m to the ground. According to his calculation it takes x seconds. He didn’t know that his fellow physicist is making a prank on him and he installed a huge magnet underneath the thin floor of the lab. Data which comes from his experiment shows that the object traveled much faster than he expected (based on the calculations made to perform this experiment, and therefore expected to prove the existence of the law of gravity). He measures the physical quantities again (checks with his fellow chemist that the object is actual iron, checks its weight again, measures the distance), he even asks his colleagues to perform the experiment independently, as he starts to question his abilities. All further experiments give the same results. Does it mean that the law of gravity is false, or all of a sudden is not holding anymore? Clearly not, they were simply not aware of an additional factor which affects his experiment – the magnetic force applied by the science prankster.

magnet

And now let’s compare this situation to the “experiments” performed by the “economists” who believe that economics is the same type of science as physics (so called quantitative economists, mostly Keynesians but also Neoclassicists). These “economists” seem to believe that performing experiments and testing hypotheses is a valid method of verifying economic laws. The first difficulty they encounter is creating an isolated, controlled and replicable environment to test a hypothesis. In contrary to physicists who work with physical objects, economists work with human objects – economic agents. For that reason, the ability to create an isolated, controlled and replicable environment in order to perform an experiment (especially macroeconomic experiments) is impossible (no not difficult, it’s just impossible). Let’s assume that some brilliant, LSE educated “economist” would like to test whether the Phillips curve model can be confirmed by experience. Thanks to the mighty European Union he has been awarded a research grant, and together with fellow “economists” in Germany, France and Spain will spend his next ten years and several million euros on collecting data on inflation and unemployment (believing that he is following a scientific method used by the physicist, remember measuring the weight of the iron object, distance and calculating the expected object travelling time). After ten of years of doing data gathering, the European consortium have gathered enough data and also… managed to confirm that indeed the Phillips curve holds true. Economic data “proved” that when inflation rises, unemployment drops, and when inflation drops, the unemployment rises. Fantastic results, the “economists” confirmed their plan, the chief economist got awarded a Nobel prize in Economics and now the central bankers around the world don’t need to rely on the ideological goals to set the monetary policy, now they can rely on “science”. Since, the Federal Reserve’s (American central bank) goal (as defined by the government) is to maintain the maximum employment, the FED’s board will now set a goal to maintain inflation rate at 2%. They do this because the experiment “has proved” that the model is true. The results of the model also showed that inflation rate of 2% corresponds to “maximum employment”. Off we go.

Ok, but what this has to do with the experiments of the physicist? Well, what the “economist” does not realize is that he is being pranked not by one prankster but by the legion. If he runs the same experiment for another ten years (remember few years have already passed since the last one)… all other independent variables which affect both the inflation and unemployment rate will change. All participating economic actors will be different. Structure and rate of taxation can be different, value of international trade will be different, legal system can be different, international commodity prices will be different, political situation can be different, demographics will be different, rate of technological progress will be different, interest rates will be different, valuation of financial assets will be different, government, corporate and household debt level will be different, velocity of money will be different, consumer sentiment will be different, period in the business cycle will be different, monetary system (fiat currencies or gold standard) can be different. All these and many more economic variables which affect both inflation and unemployment will be different, and so they will be playing a prank on the “economist” in the same way as his fellow “magnet would be fun” scientist did.

stock-trading

And this difficulty exists not only to “staged” experiments, which “the economist” would like to perform in the future (which he can’t as he can’t just fix all these variables on the same level as they were in the first experiment, therefore he can’t set an isolated, controlled and replicable environment which is the foundation of hypothesis testing in physics). This same difficulty also applies to the hypothesis testing based on historical data (as a matter of fact the first example is also based on historical data, as the experiment can only be finalized once all data has been collected, therefore it has already became a historical data). Let’s assume an economist undertook an ambitious task of running Phillips curve hypothesis testing on 10 different, 10-years period between 1900 and 2000. The first experiment takes inflation and unemployment between 1900 and 1910, the second between 1910 and 1920, and so on. Clearly none of these periods will be the same (and not even similar) in regards to the values of the previously mentioned economic variables which affect both dependent variable (inflation and unemployment rate). Difference between values of the prank variables across time can sometimes be small and sometimes can be huge. Between 1900 and 2000, monetary system which affects all business transactions in the economy has changed few times – from international gold standard, to fiat currency, to gold exchange standard, to fiat currency. Economic actors in all these different systems are exposed to different incentives, therefore their economic actions (which will affect inflation and unemployment rate) will be radically different. Monetary system based on gold is a deflationary system, where people have economic incentives to save money (as money naturally gains purchasing power), while fiat system is an inflationary system where people have economic incentives to spend and borrow money (as money loses purchasing power due to government and fractional reserve banking increase in money supply). Even if some “economist” was aware of these difficulties and did all he could to compare some periods which are “similar”, this could not be achieved as the economic variables are never fix. The only constant in the realm of economic variables is change. Every second variables change, and every second the world as described by economic data is different.

For that reason, it’s simply not possible to run any kind of hypothesis testing or experiments in the realms of economics with the assumption that the experiments are done through the same scientific method which is used in physics. Scientific method of natural science is useless in economics. “Economists” are pranked not by one fellow, they are fooled by the legion pranksters.

prankster

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