# Mathematical Probability in Science

It is perfectly acceptable in thought to characterize material processes as mathematically random. For example, the roll of two dice is characterized as random such that their sum of seven is said to have a probability of 1/6. The equation of radioactive decay may be characterized as the probability function of the decay of a radioactive element. Wave equations in quantum mechanics may be characterized as probability functions. However, such valid mathematical characterizations do not attest to randomness and probability as being characteristics of material reality. Rather, such characterizations attest to mathematical randomness and probability as being characteristic of human knowledge in its limitations.

If randomness and probability were characteristic of material reality at any level, including the atomic and sub-atomic level, material reality would be inherently unintelligible in itself. Material reality would be inexplicable and as such inherently mysterious. Yet, to view material reality as inherently mysterious is superstition. Superstition denies causality by claiming that material results are a mystery in themselves, e.g. that they are materially random.

It is an erroneous interpretation to hold that quantum mechanics requires material reality to be random and probable in itself. Wave equations may be viewed as probability functions only in the same sense that the result of rolling dice is mathematically probable. That sense is in the suspension of human knowledge of material causality at the level of physical forces for the sake of a mathematical utility without the denial of material causality.

A commenter on a recent post at catholicstand.com (ref. 1), was so enamored with the validity, utility and beauty of the mathematics of quantum mechanics that he declared, “This randomness is inherent in nature.” Indeed it is inherent in nature, i.e. in human nature in the limitations of the human knowledge of the measurable properties of material reality.

Material reality is not random in its nature. The nature of material reality, in light of the utility of application of the mathematics of probability or in light of perceiving a mathematical function as one of probability, is not a question within the scope of science or mathematics. The nature of material reality is always a question within philosophy. In contrast, the mathematical and scientific question is the suitability of specific mathematics in representing the relationships among the measurable properties of material reality including those properties, which can only be detected and measured with instruments.

Let it be noted that scientific knowledge cannot demonstrate the fundamental invalidity of human sensory knowledge and human intellectual knowledge because the validity of scientific knowledge depends on the fundamental validity of these.

It has been recognized since the time of Aristotle that the human intellect is extrinsically dependent in its activity upon a sensual phantasm, i.e. a composite of sense knowledge. This and all visualizations or imaginative representations are necessarily restricted to the scope of the senses, although the intellect is not. Consequently, science at the atomic and sub-atomic level cannot consist in an analysis of visual or imaginative simulations, which are confined to the scope of human sensation. Rather, the science consists in the mathematics, which identifies quantitative relationships among instrumental measurements. It would be a fool’s quest to attempt to determine a one to one correspondence between science and an imaginative representation of the atomic and sub-atomic level or to constrain the understanding of the science to such a representation (Ref. 2).

Remarkably, in an analogy of a wave function in quantum mechanics as a probability function which collapses into a quantum result, the physicist, Stephan M. Barr, did not choose an example of mathematical probability (Ref. 3). He could have proposed an analogy of mathematical probability simulated by flipping a coin. When the coin is rotating in the air due to being flipped it could be viewed as a probability function of heads of 50%, which collapses into a quantum result of heads, namely one, or tails, namely zero, upon coming to rest on the ground.

Instead, he chose an example where the meaning of probability is not mathematical, but qualitative.

Mathematical probability is the fractional concentration of an element in a logical set, e.g. heads has a fractional concentration of 50% in the logical set of two logical elements with the nominal identities of heads and tails. A coin is material simile.

A completely unrelated meaning of the word, probability, is an individual’s personal lack of certitude of the truth of a statement. Examples: ‘I probably had eggs for breakfast in the past two weeks’ or ‘Jane will probably pass the French exam.’ These statements identify no set of elements or anything quantitative. Personal human certitude is qualitative. Yet, we are bent upon quantitatively rating the certitude with which we hold our personal judgments.

Barr succumbs to this penchant for quantifying personal certitude. He illustrates the collapse of a wave function in quantum mechanics with the seemingly objective quantitative statement:
“This is where the problem begins. It is a paradoxical (but entirely logical) fact that a probability only makes sense if it is the probability of something definite. For example, to say that Jane has a 70% chance of passing the French exam only means something if at some point she takes the exam and gets a definite grade. At that point, the probability of her passing no longer remains 70%, but suddenly jumps to 100% (if she passes) or 0% (if she fails). In other words, probabilities of events that lie in between 0 and 100% must at some point jump to 0 or 100% or else they meant nothing in the first place.”
Barr mistakenly thinks that probability, whether referring either to mathematics or to human certitude, refers to coming into existence, to happening. In fact, both meanings are purely static. The one refers to the composition of mathematical sets, although its jargon may imply occurrence or outcome. The other refers to one’s opinion of the truth of a statement, which may be predictive. That Jane has a 70% chance or will probably pass the French exam obviously expresses the certitude of some human’s opinion, which has no objective measurement even if arrived at by some arbitrary algorithm.

Probability in mathematics is quantitative, but static. It is the fractional composition of logical sets. Probability in the sense of human certitude, like justice, is a quality. It cannot be measured because it is not material. This, however, does not diminish our penchant for quantifying everything (Ref. 4).

Barr’s identification of probability, as potential prior to its transition to actuality in an outcome, is due to taking the jargon of the mathematics of sets for the mathematics of sets itself. We say that the outcome of flipping a coin had a probability of 50% heads prior to flipping, which results in an outcome or actuality of 100% or 0%. What we mean to illustrate by such a material simulation is a purely static relationship involving the fractional concentration of the elements of logical sets. The result of the coin flip illustrates the formation or definition of a population of new sets of elements based on a source set of elements. In this case the source set is a set of two elements of different IDs. The newly defined population of sets consists of one set identical to the original set, or, if you wish, a population of any multiple of such sets.

Another illustration is defining a population of sets of three elements each, based on the probabilities of a source set of two elements of different nominal IDs, such as A and B. The population is identified by eight sets. One set is a set of three elements, A, at a probability (fractional concentration) of 12.5% in the newly defined population of sets. One set is a set of three elements, B, at a probability of 12.5%. Three sets are of two elements A and one element B, at a probability of 37.5%. Three sets are of two elements B and one element A, at a probability of 37.5%. The relationships are purely static. We may imagine the sets as being built by flipping a coin. Indeed, we use such jargon in discussing the mathematics of the relationship of sets. The flipping of a coin in the ‘building’ of the population of eight sets, or multiples thereof, is a material simulation of the purely logical concept of random selection. Random selection is the algorithm for defining the fractional concentrations of the population of eight new sets based on the probabilities of the source set. It is only jargon, satisfying to our sensual imagination, in which the definitions of the eight new sets in terms of the fractional concentration of their elements are viewed as involving a transition from potency to act or probability to outcome. The mathematics, in contrast to the analogical imaginative aid, is the logic of static, quantitative relationships among the source set and the defined population of eight new sets.

Random selection, or random mutation, is not a material process. It is a logical concept within an algorithm, which defines a logical population of sets based on the probabilities of a logical source set.

It is a serious error to conflate mathematical probability with the certitude of human judgment. It is also a serious error to believe that either refers to coming into existence or to the transition from potency to act, which are subjects of philosophical inquiry.

Ref. 1 “When Randomness Becomes Superstition” http://catholicstand.com/randomness-becomes-superstition/

Ref. 2 “Random or Non-random, Math Option or Natural Principle?” https://theyhavenowine.wordpress.com/2014/08/24/random-or-non-random-math-option-or-natural-principle/

Ref. 3 “Does Quantum Physics Make It Easier to Believe in God?” https://www.bigquestionsonline.com/content/does-quantum-physics-make-it-easier-believe-god

Ref. 4 “The Love of Quantification” https://theyhavenowine.wordpress.com/2013/08/11/the-love-of-quantification-2/