True human love is real. It is the will to self-sacrifice. In contrast, falling in love is falling for make believe. Somewhat similarly we must be on guard that our love of quantification is not a falling in love, which blinds us to the distinction between material reality and the logic of pure mathematics.
The physical sciences aim to discover mathematical relationships inherent in the measurable properties of material reality. The experimental determination, that for any object the ratio of an unbalanced force to the consequent acceleration is a constant, leads to naming the constant, mass. Similarly, if the ratio of voltage to electrical current is constant for some material, then the constant is named the resistance of the material to which the voltage is applied. These mathematical relationships are discovered by experimentation.
Not long ago I saw a TV ad in which a five year old lad at play was maneuvering stones with a stick. As he grows older, the lad would be bored to death at such a game, except for its quantification. With quantification, such play becomes absorbing and its participation and enjoyment last a lifetime. Through quantification the stick and object are geometrically defined in various games as is the geometry of the play lot. With the addition of a few other quantitative rules, we have croquet, badminton, hockey, golf, baseball, tennis, etc., even bowling in which the stick is our arm. The quantitative relationships of games are not discovered, as the mathematical relationships of science are discovered. They are agreed upon by convention.
We have such a love of quantification that we even like to quantify that which is not measurable. This includes justice and fortitude and for this essay, human certitude. Often in discussion shows, the pundits are asked to quantify the certitude of their opinions on a scale of 0 to 10.
However, it is not simply the love of quantification which impels us to quantify human certitude. The character of experimentation requires it as a convention based utility. The need for a utility in quantifying certitude is not obvious in physics and chemistry, but it is in biology. In biology the inherent variability of the experimental material makes it difficult to distinguish between variation due to the manipulated experimental variables and the uncontrolled variation inherent in the experimental material itself. The solution has been to design experiments in such a way as to allow the distinction between variation due to the manipulated variables and variation identified as experimental error, i.e. uncontrolled variation, largely due to the experimental material. Along with assumptions of how repeated measurements vary about a mean, a universally accepted convention of assigning quantitative values to the human certitude of repetitive measurements has been established.
The mathematics of such conventions has been named, statistics. Conventions for assigning quantitative values to human certitude have also been applied to the prudence of decision making and to the art of prediction.
The lack of full certitude of the truth of a proposition is often described using the word, probability. Note that probability does not characterize reality. It characterizes human knowledge and, more accurately, human ignorance. The focus of statistics is establishing a convention for quantifying human certitude based on data. However, probability, with an entirely different definition, which has nothing to do with human certitude or with data, is also a topic in mathematics. Yet, statistics heavily depends upon the mathematics of probability.
Perhaps the most important aspect of the mathematics of probability is that it is not applicable to reality except by the analogy of emulation. The mathematics of probability is a purely logical exercise because one of its foundational premises cannot be true of material reality.
In mathematics, probability is the fractional concentration of an element in a logical set. The mathematics of probability is the elaboration of the relationships of logical elements and logical sets primarily through the formation of new sets based upon the probabilities of a source set. This process is called random selection. In random selection all that matters is the fractional concentrations of the elements in the source set. In other words, the only relevant characteristic of an element in a set is its ability to be counted. The IDs of the elements in the sets are purely nominal. In contrast, nothing material is purely nominal by having only one characteristic, namely the fact that it can be counted. (Note: Under other topics in mathematics, not probability, the fractional concentration of an element in a set can be treated as a density.)
In the mathematics of probability considerable leeway is allowed in the use of jargon. One would not ask, “What is the probability of an element of a set of one element formed by random selection from a source set consisting of four subsets, where each subset consists of thirteen ranked elements?” More likely one would use the jargon, “What is the probability of drawing the three of diamonds from a deck of cards? Notice that in spite of the jargon, the question is one in pure mathematics. The question is not, “How good, as an emulation of a probability of 1/52 in characterizing newly formed sets of one, would be the shuffling of a deck of cards and drawing the top card?”
It is perfectly legitimate to use jargon in the discussion of the purely logical relationships of the mathematics of probability, where the jargon uses visual counterparts to the purely logical entities. In the above jargon, the logical source set of 52 unique elements has for its visual counterpart a deck of cards. Random selection has for its visual counterpart the shuffling of the deck. The newly formed logical set has the top card as its visual counterpart.
For me, the following use of jargon is an excellent illustration of just how purely logical is the mathematics of probability. This purely logical character is evident in the impossibility of a full correspondence to visual counterparts in the illustration: “What is the probability of any sequence of a deck of playing cards?” In this use of jargon, the shuffling of the deck would be the counterpart of random selection, while the shuffled deck would be the single element in the newly formed set. However, it is humanly impossible to visualize a counterpart of the source set, whose elements would be one deck of every possible sequence. For homework, what would be the mass of such a source set, if each deck of cards in the source set had the mass of an atom of hydrogen? This source set would consist of 8.0 x 10^67, i.e. 52 factorial elements. There are 6 x 10^23 atoms of hydrogen per gram. The mass of the earth is 6 x 10^24kg.
In reality, random selection is a material impossibility, because no material thing can be materially selected without respect to its material properties, including location. In material analogies of random selection, human ignorance is equated with randomness. The material emulation of the mathematics of probability requires the suspension of the human knowledge of material causality at the level at which randomness is posited. To attribute randomness to material reality would be to deny the inherent intelligibility of material reality and with that the possibility of all human knowledge and communication.
The two self-evident principles upon which all human knowledge and communication depend are that things exist and that everything makes sense, i.e. material reality is inherently intelligible. A self-evident principle is one, if denied, renders impossible all human knowledge and communication. Randomness outside of pure logic, would be a denial of the principle that everything makes sense.
We make severe errors, if we think that the discussion of mathematical probability has any direct correspondence to material reality as do the mathematical relationships discovered as inherent in the measureable properties of material reality. We make severe errors, if we think that mathematical probability has anything to do with existence or coming into existence. The mathematics of probability is purely logical.
The logical concept of random selection from a source set is synonymous with the logical concept of the random mutation of the elements of a source set. When the source set consists of n unique elements, then the logical concepts of random selection and random mutation are synonyms of the logical concept of random numbers generation to the base, n. Although random selection, random mutation and random numbers generation allude to motion, the mathematics of probability is one of statics. The only “motion” in the mathematics of probability is the formation, i.e. the definition, of new logical sets based on the probabilities of a source set. There can be no correspondence inherent in material reality to definitions, which are logically formative of sets based on randomness. The only correspondence is one of emulative analogy based on the suspension of human knowledge of the material forces operative at the level at which randomness is posited. The mathematics of probability may be employed as a utility to compensate for human ignorance or to estimate human ignorance based on a convention of assumptions. The mathematics may also be employed in games were the emulation of randomness is conceded by the players for the sake of the game.
In the mathematics of probability, the jargon of material emulation is employed for the sake of convenience. In statistics, mathematical probability is employed in the establishment of conventions. In the toss of a coin before a football game and in the concoction of games such as poker, mathematical probability is employed by concession of the players.
We all love quantification. However, no one should fall in love with quantification, which results in blindness to reality by mistaking purely logical entities and relationships for reality. One sign of having fallen in love with quantification is the error of believing that mathematical probability can be inferred from material reality. For one particular discussion of such impossible inference, see “What is modern in the new atheism? – the inference of probability” (Part 1, http://deltaepsilonsigma.org/media/delta-epsilon-sigma/1DES_Journal_FALL12F.pdf and Part 2, http://deltaepsilonsigma.org/media/delta-epsilon-sigma/Volume-LVIII-Spring2013-Number1.pdf)