I mentioned I
had difficulty following some of the mathematical theorems taught there
although I scored 4 A1s for mathematics and science when I left school.
My
brother-in-law replied that was “those days” which I thought he may mean
mathematics is easier “these days”
Let me reply
to him how tough mathematics can be as Queen of Science. It was Carl Friedrich
Gauss, the famous mathematician who said that mathematics is ‘the queen of
sciences’ just like we think Physiology is the Queen of Medicine
Let me explain
my difficulty studying some branches of mathematics at university degree level
whether it was “those days” or “now”.
The difficulty
of different branches of mathematics can be subjective and varies from person
to person, depending on their interests, strengths, and experiences like me.
However, some branches are generally considered more challenging than others.
I read
mathematics and physics as one of my undergraduate degrees at the Muslim
University of Aligarh in the 1960’s and again some mathematics when I did a
postdoctoral course in astronomy at the University of Oxford long after my
retirement in medical research and just before the Covid pandemic. I read it as
a hobby, not for a job.
Here’s
my personal view and experience with mathematics and physics.
The easiest
branch of mathematics everyone studies in school is arithmetic that deals with
basic number operations like addition, subtraction, multiplication, and
division.
This of course
is the first area of math taught, very concrete and intuitive before going to
basic algebra that involves solving simple equations and understanding basic
algebraic expressions. Simple algebra builds directly on arithmetic and
introduces the use of symbols and simple equations.
Geometry
(Euclidean) studied in schools are properties and relationships of points,
lines, surfaces, and shapes in two and three dimensions. Often visual and
intuitive, dealing with concepts that can be easily visualized. Geometry is
also very simple to understand.
Higher
understanding of mathematics studied at university level probably is algebraic
geometry. This branch of mathematics studies solutions to algebraic
equations and their properties using both algebra and geometry. This requires a
deep understanding of abstract algebra, complex geometry, and topology. It
involves intricate and abstract concepts. This involves topology that studies
the properties of space that are preserved under continuous transformations.
This is a highly abstract mathematics dealing with properties and structures
that may not have direct physical interpretations. Concepts like homotropy,
homology, and topological spaces can be very difficult to grasp.
Then we also
have this thing called analytic number theory that uses mathematical analysis
to solve problems about integers. This is very challenging as it involves
complex and advanced techniques from analysis and algebra. Famous problems like
the Riemann Hypothesis belong to this area.
Abstract
algebra is another area that involves algebraic structures such as groups,
rings, and fields. This branch of mathematics deals with very abstract concepts
that require strong logical reasoning and the ability to handle non-intuitive
structures.
Next, we also
have to deal with differential geometry that uses calculus and algebra to study
problems in geometry. This branch requires a good grasp of both advanced
calculus and abstract algebra, dealing with complex and abstract geometric
structures.
Statistics is
a branch of mathematics we as medical researchers use extensively for
epidemiological studies, designing medical and clinical studies and the
analysis of the data we collect and interpret. In short, this involves the
collection, analysis, interpretation, presentation, and organization of data.
Statistics with lots of equations and formulae can be very tough to understand
but reachable to us who have a very strong mathematical background.
This is also
quite accessible to other biomedical scientists with strong mathematical
background, normally they do, though quite a challenge for clinicians
(medical doctors) to understand. Statistics, I should say, is practical and
applied, often considered easier due to its direct applications and use of
intuitive concepts like mean, median, and mode.
Then we have
basic calculus studied in schools as “additional mathematics.” Basic calculus
introduces concepts of limits, derivatives, and integrals. While it can be
challenging initially for ordinary students, basic calculus is generally
manageable and provides powerful tools for solving practical problems such as
volumes of odd shapes and designs, and continuous motions or accelerations or
on gravitation.
The perceived
difficulty of a mathematical branch can vary greatly among individuals. Some
may find abstract branches like topology and algebraic geometry particularly
challenging, while others might struggle with more applied areas like
statistics or calculus. The key to mastering any branch of mathematics lies in
practice, persistence, and finding resources that suit one's learning style.
Then what
about tensor calculus or mathematics used in astrophysics or particle physics
for Standard Model?
Wow! This
branch of mathematics requires almost a mathematical genius like Albert
Einstein to understand. Tensor calculus is a branch of mathematics that extends
the concepts of vectors and matrices to more complex structures called tensors.
It's a key mathematical tool in fields like differential geometry, general
relativity, and continuum mechanics that Einstein used. Its complexity is very
high. This branch of mathematics requires almost the brain of a genius I should
say. It requires a solid understanding of linear algebra, differential
geometry, and multivariable calculus.
Tensor
calculus is used to describe physical laws in a form that is independent of the
choice of coordinates, making it crucial for the formulation of Einstein's
field equations in general relativity. That’s tough indeed.
What about the
mathematics we used in astrophysics? Wow! That’s another area not meant for
ordinary mathematicians or physicists I should say.
Mathematics we
use in astrophysics often includes differential equations, where both ordinary
and partial differential equations are used to model physical systems, such as
the evolution of stars, the dynamics of galaxies, and the behaviour of fluids
in astrophysical contexts.
Other branches
of mathematics that can crack our heads are numerical methods that are
essential for solving complex equations that cannot be solved analytically,
such as N-body simulations for galaxy dynamics or solving the Navier-Stokes
equations for astrophysical fluid dynamics. An example is Fourier Analysis
where we use in signal processing for analysing data from telescopes and other
instruments. We use this branch of mathematics in astronomy, especially in
radio astronomy. It involves statistical methods that are important for data
analysis and interpretation, including Bayesian statistics for parameter
estimation and model comparison.
Its complexity
is high since astrophysics requires a broad range of mathematical tools and the
ability to apply them to physical problems.
Next, we also
have to deal with mathematics used in Particle Physics (Standard Model). This
requires Group Theory for the understanding of the symmetries of particles and
their interactions. The Standard Model is based on the gauge group such
as:
SU (3) c
x SU (2) l x U (I) y
One of the
subjects in both mathematics and physics is Quantum Field Theory (QFT). Here we
combine quantum mechanics with special relativity to describe the behaviour of
subatomic particles. This includes the use of Lagrangians, path integrals, and
Feynman diagrams.
Renormalization
is a process used to handle infinities that arise in QFT calculations.
Whereas
Representation Theory is used to classify particles, and their interactions
based on their properties under the gauge groups of the Standard Model.
The complexity
of this branch of physics and its mathematics I admit is extremely high as it
requires a deep understanding of advanced mathematical concepts and their
physical interpretations.
Let me
summarise to make mathematics easier to understand for non-mathematicians.
First, tensor
calculus and mathematics used in astrophysics and particle physics are among
the most challenging areas in mathematics due to their high level of
abstraction and the depth of understanding required.
Second, tensor
calculus is crucial for theoretical physics, particularly for general
relativity.
Third, in
astrophysics, this involves a wide range of mathematical techniques, often
applied to complex and large-scale physical systems.
Fourth,
particle physics (Standard Model) relies heavily on advanced mathematical
structures like group theory and quantum field theory.
These areas
represent the cutting edge of both mathematics and physics, requiring not only
a strong foundation in various mathematical disciplines but also the ability to
apply these concepts to solve intricate and abstract problems.
Having
explained all these, do you think you can study mathematics and physics? Better
not. Mathematics is not for everyone. It is reserved only for mathematical
genius and wizards like Albert Einstein.
Mathematics is
definitely not an ABC subject like biology, medicine or nutrition for sure –
these areas I too have studied at postgraduate levels plus mathematics at
undergraduate degree level.
Take care with
STEM (Science, Technology, Engineering and Mathematics). They are not meant for
everybody.
ju-boo lim
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