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 problemsolving, critical thinking, language, and more
by clicking here.
Much of Math is problemsolving. 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 nontrivial (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.
© 2023. All rights reserved.
If you have any comments, concerns, critiques, or suggestions I
can be reached at mwd@profitpages.com.
I will review reasoned and intellectual correspondence, and it is
possible that I can change my mind,
or at least update the content of this article.
