The Personal Website of Mark W. Dawson

Containing His Articles, Observations, Thoughts, Meanderings,
and some would say Wisdom (and some would say not).

Math (and Statistical Mathematics) is More than Numbers

On the surface, mathematics may seem like it is all about numbers and formulas. However, this versatile subject is about much more than just arithmetic, algebra, and trigonometry (as is generally taught in high school). Discover why math is more than numbers, and find out how it contributes to the development of valuable skills in problem-solving, critical thinking, language, and more by clicking here.

Much of Math is problem-solving. Solving the problem, then proving your solution is correct, is what most mathematicians do. However, no solution is totally provable, and many problems are unsolved. Some examples of this are:

Gödel's incompleteness theorems deal with a wide class of logical systems that cannot be both consistent and complete. Gödel's incompleteness theorems are the name given to two theorems (true mathematical statements), proved by Kurt Gödel in 1931, and they are theorems in the field of mathematical logic.

Mathematicians once thought that everything that is true has a mathematical proof. A system that has this property is called complete; one that does not is called incomplete. Also, mathematical ideas should not have contradictions. This means that they should not be true and false at the same time. A system that does not include contradictions is called consistent. These systems are based on sets of axioms, which are statements that are accepted as true and need no proof.

Gödel said that every non-trivial (interesting) formal system is either incomplete or inconsistent:

• There will always be questions that cannot be answered using a certain set of axioms;
• You cannot prove that a system of axioms is consistent unless you use a different set of axioms.

Those theorems are important to mathematicians because they prove that it is impossible to create a set of axioms that explains everything in math.

Hilbert's problems are 23 problems in mathematics published in 1900 by mathematician David Hilbert. This list of problems turned out to be very influential. After Hilbert's death, another problem was found in his writings; this is sometimes known as Hilbert's 24th problem today. This problem is about finding criteria to show that a solution to a problem is the simplest possible. Of the 23 problems, three were unresolved in 2012, three were too vague to be resolved, and six could be partially solved. Given the influence of the problems, the Clay Mathematics Institute formulated a similar list, called the Millennium Prize Problems, in 2000.

Mathematics (and statistical mathematics) cannot solve every problem. Some problems have so many constants and variables as to be unsolvable. And as one of Murphy’s Laws states, Variables won't, and constants aren't, which makes solutions difficult to obtain. There is also the problem of what we know, what we don’t know, and what we don’t know that we don’t know, as discussed in my Article “Oh What a Tangled Web We Weave”. Therefore, keep in mind when someone (even an expert) utilizes mathematics or statistics to prove something, they are more probably wrong than they are probably right, especially in the use of math or statistics in regard to social policy (for more about utilizing statistics within social policy I would recommend the book “Discrimination and Disparities” by Thomas Sowell).

Also, good science requires good mathematics. But mathematics is abstract, while science is based on reality. As such, mathematics is a contributor to science and not a substitution for science, as I have Chirped on "03/30/21 Mathematics is Not Science". Therefore, be wary when someone claims that mathematics proves the science. Science is proved or disproved by observations and experiments, not math. A good book that examines this issue is “Lost in Math: How Beauty Leads Physics Astray” by Sabine Hossenfelder.

Mathematics is very powerful, and a good mathematician can be very valuable in obtaining solid solutions to problems. However, incorrect mathematics and mistaken mathematicians can lead you astray and to an improper solution. Consequently, mathematics should be utilized with care and never taken at face value, especially when non- mathematicians are utilizing mathematics.

Disclaimer

Please Note - many academics, scientist and engineers would critique what I have written here as not accurate nor through. I freely acknowledge that these critiques are correct. It was not my intentions to be accurate or through, as I am not qualified to give an accurate nor through description. My intention was to be understandable to a layperson so that they can grasp the concepts. Academics, scientists, and engineers entire education and training is based on accuracy and thoroughness, and as such, they strive for this accuracy and thoroughness. I believe it is essential for all laypersons to grasp the concepts of this paper, so they make more informed decisions on those areas of human endeavors that deal with this subject. As such, I did not strive for accuracy and thoroughness, only understandability.

Most academics, scientist, and engineers when speaking or writing for the general public (and many science writers as well) strive to be understandable to the general public. However, they often fall short on the understandability because of their commitment to accuracy and thoroughness, as well as some audience awareness factors. Their two biggest problems are accuracy and the audience knowledge of the topic.

Accuracy is a problem because academics, scientist, engineers and science writers are loath to be inaccurate. This is because they want the audience to obtain the correct information, and the possible negative repercussions amongst their colleagues and the scientific community at large if they are inaccurate. However, because modern science is complex this accuracy can, and often, leads to confusion amongst the audience.

The audience knowledge of the topic is important as most modern science is complex, with its own words, terminology, and basic concepts the audience is unfamiliar with, or they misinterpret. The audience becomes confused (even while smiling and lauding the academics, scientists, engineers or science writer), and the audience does not achieve understandability. Many times, the academics, scientists, engineers or science writer utilizes the scientific disciplines own words, terminology, and basic concepts without realizing the audience misinterpretations, or has no comprehension of these items.

It is for this reason that I place understandability as the highest priority in my writing, and I am willing to sacrifice accuracy and thoroughness to achieve understandability. There are many books, websites, and videos available that are more accurate and through. The subchapter on “Further Readings” also contains books on various subjects that can provide more accurate and thorough information. I leave it to the reader to decide if they want more accurate or through information and to seek out these books, websites, and videos for this information.