What exactly does mathematics give you when you apply it to the study of some phenomenon, process or observation?
Without mathematics there is no way you can understand what keeps an airplane in the air. As we all know, metal objects don't stay above the ground without something to support them. But when you look at a jet aircraft flying overhead, you can't see anything holding it up. It takes mathematics to 'see' what keep an airplane aloft. In this case, what let you see the invisible is an equation (Bernoulli's equation) discovered by the mathematician Daniel Bernoulli early in the eighteenth century. Bernoulli's equation was established by Bernoulli over a century before the first man-made wings were used for the purpose of flight.
Another example, what is it that causes objects other than airplanes to fall to the ground when we release them? The immediate answer is "Gravity". But that is just giving it a name; it does not help us to understand it, it is still invisible. To understand gravity we have to 'see' it and that is exactly what Newton did with his equations of motions and mechanics in the seventeens century. Newton's mathematics enabled us to 'see' the invisible forces that keep the earth rotating around the sun and cause an apple to fall from a tree onto the ground.
From the above just two examples, we can see that one answer to the above question is that mathematics makes the invisible visible.
There may be more than one answer to this question. In your opinion what does mathematics do?
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This question was first considered by Keith Devlin in his book: "The language of Mathematics Making the invisible visible". The above examples are just two of many considered by the author.
What make an airplane floats are the shapes of objects that are used to create the plane and the forces involved against gravity that enable the plane to levitate - levity as opposed to gravity. For these combinations of things to function properly , mathematics plays a fundamental role - to precisely simulate these forces and describe them through differential equations and enable the engineers to create a design based on the mathematical formulations assumed. Mathematics therefore is an enabler that makes us understand forces of nature in particular and natural phenomena in general to look in to, understand and communicate with it. It is therefore mathematics is the universal language (or I can say metalanguage ) of the greater universe.
Well, back in my first University times (I studied physics at that time), I remember our building was in front of (say actually opposed to) Mathematics. And we had nice and friendly but long discussions on a similar topic, well related to your question. Mathematicians there said Mathematics is a science per se (let's interprete it was discovered), Physicists said Mathematics are a set of tools to compute what is observed in "real science" (rudely translated from spanish), so let's interprete is was invented. I think this is more depending on the point of view rather than on the truth. After the years, I actually now think also physics were invented (the things we observe are like they are, but the models to explain them are definitely invented by us, and this models are essentially expressed through maths). Further, a professor of my third University times (I studied Visual Sciences at this time), always defined Mathematics not even as science or set of tools, but just as a "convenient language to rigorously express what has been observed" Do not know whether it helps...
Thank you Samuel. The question is not about if mathematics is invented or discovered. Your professor is right in saying that mathematics rigorously express what has been observed but that is not the full story. In expressing what has been observed mathematics makes predictions and can reveal hidden phenomena or process.
Let me give you an example. I am an astrophysicist and in my field of research I was trying to understand the structure, stability and evolution of complex type of binary stars called contact binary stars. When observational data is put into a mathematical form the equations tell us that the primary (more massive) star is under-luminous for its mass and the secondary (less massive star) is over-luminous for its mass. Since energy is conserved this lead to the inescapable conclusion that these stars are somehow sharing their nuclear generated energies with the more massive star transferring energy to the less massive star. In other words, when observational data was put into mathematical form, the equations revealed a hidden phenomenon, energy transfer is taking place inside the common structure of these stars. Mathematics enabled us to 'see' an invisible process that is taking place millions of miles away deep inside the structure of these contact binary stars.This conclusion has a profound influence on our understanding of these type of stars structure, stability and evolution. This is yet another example of mathematics making the invisible visible.
There are many such examples in science. But what I am also looking for is an answer that goes beyond this. Perhaps somebody can shed more light on this from his/her own research experience.
On a separate note, I totally disagree with your professor on the issue that mathematics is not science, but that is a separate issue not related to the question of this thread and shall not discuss it further.
Thank you Issam,
but in your example is is not mathematics telling you that "When observational data is put into a mathematical form the equations tell us that the primary (more massive) star is under-luminous for its mass and the secondary (less massive star) is over-luminous for its mass". This is derived from interpretation of the results of applying a physical model to your observations.
Of course, you NEED maths for describing and applying the model rigorously but the model goes beyond maths.
So I still disagree that maths is responsible for the discovery, maths is the most elegant and convenient language to express physics, nothing more and nothing less, but that's enough.
and I love maths...
Thank you Samuel,
No. This is derived from comparing the M-L relation of the contact system (derived from observation) to the M-L relation to stars as single stars. Physics tells us that contact alone should not alter the nuclear energy generated in the center of each star as single star. I am comparing mathematical relations derived from observations of two stars in contact to a the relations of two stars (of the same mass) as single stars. It is the observed mathematical relations that lead us to the conclusion I explained earlier.
A physical Model can't be built without maths so how can you say it goes beyond maths? A physical model can't go anywhere without math!
Exactly, the physical model includes maths but also the concepts that are then linked through maths, so yes i still believe physics goes beyond maths, but physics needs maths to express itself
The IRMS 2010 expresses the importance of mathematical sciences in the following manner:
“the mathematical sciences provide a universal language for expressing abstractions in science, engineering, industry and medicine; mathematical ideas, even the most theoretical, can be useful or enlightening in unexpected ways, sometimes several decades after their appearance; the mathematical sciences play a central role in solving problems from every imaginable application domain; and, because of the unity of the mathematical sciences, advances in every sub-area enrich the entire field.”
In additon, mathematical science is also a hugely important discipline in its own right, with its own culture and intellectual imperatives, its own history over millennia, and its own ‘Grand Challenges’. It is important to see mathematics in its entirety and not be distracted by the misleading distinction between theory and applications, often expressed as ‘pure’ versus ‘applied’ mathematics.
Finally, is it also important to recognize that mathematical sciences provide the tools and the appropriate models to study all kind on natural and social phenomena.
"Mathematics is queen of sciences" this one line gives us all idea about Mathematics. Mathematics is a tool for progress/ inventions that we have today. If we see properly , Mathematics is everywhere around us. We are actually surrounded by matjematics. Mathematics is language of science. Without mathematics , no science will survive. If you see any ckt is a differential equation. Every network is either a topology or a system of equations. Every material has its own mathematics. We need to see the mathematics behind every situation that happens in the world then we wil understand what mathematics have to do with life?
I would say mathematics is about logical abstractions. It condenses all logical observations into abstractions, which are free of any instances. Let's take linear algebra as an example of a mathematical theory; the axioms of linear space makes it possible to test if some instance, such as displacements or rotations of rigid objects in space, fulfil the required basic properties. And if the axioms are satisfied, then we may exploit the whole theory, in this case linear algebra, to predict all kind of properties to such objects. Hence the prediction power of mathematics into physics and engineering.
It's another philosophical issue, why physics involves, or as some people would say, why physics 'obeys', such logical abstractions? (As a side note let me add, in my understanding, the nature does not obey physics, but instead physics is, like mathematics, a model developed by the mankind. Consequently, the nature does not obey physics, but it's the other way around, the model of physics points to what we observe in the nature.) In my current understanding this philosophical issue boils down to the fact that when building physics (i.e., the model) we as the mankind have assume "the nature appears to us in the same way under the same conditions^*". This assumption seems to involve a statement of logic that affects the whole model construction.
^* A simple example from classical physics; Gravitation is part of mechanics. Then we discover that in some cases, such as when combing dry hair, something odd happens in the sense of gravity. According to the basic assumption of physics, we still do not come up with the idea gravitation were somehow different in case of a comb and hair. But instead, we rather conclude the conditions are different. Hence, although the question is of forces, we find a need to explain this observation in terms of a new phenomenon, and this then leads to what is called by the name electricity (or in general, electromagnetism).
@Martin Kovar: ...mathematics is a very general way how to study, discover and capture the structure of the Nature or the Universe.
I agree. You may want to consider taking your idea a step further, since we can capture structures in nature by introducing what you have called a framework and a framology for a particular set of places (non-abstract points). Then, for example, we may want to consider a Smirnov continuous mapping f on a framology X to a framology Y. To do this, let X and Y be endowed with an Efremovic proximity relation. Let x be an adherent point in X. Then f is continuous, provided f(x) is adherent in Y. This approach to proximally continuous mappings between framologies will work, provided we define a descriptive adherent point x in X to be a non-abstract point whose description does not match the description of any other point in X. Then, for a descriptively adherent point in X, a mapping on X to Y is proximally continuous, provided f(x) is a descriptively adherent point in Y.
Consider any set of non-abstract points (places having locations and measurable content). We can define a proximal framework.
Call the set of places X and for measurable content, consider extracting feature values from the places in X. Let $\Phi$ be a set of probe functions used to extract feature values from each place x in X. Now endow X with a descriptive proximity relation $\delta_{Phi}$. Let $\Phi(x)$ be a feature vector that describes x. And let X be endowed with a metric d. This means that given any two sets A and B in X, the places are near, provided there is at least one a in B and b in B such that $d(\Phi(a),\Phi(b)) = 0$. Otherwise, a and b have dissimilar descriptions. Notice that sets can be descriptively near and yet have empty intersection (no points in common, spatially). In other words, disjoint sets of places can be descriptively near.
A point a in A is descriptively adherent in A, provided the description of any y in A\{a} does not match the description of a. This is quite different from the usual interpretation of point of adherence of a set.
Now take this approach a step further to obtain a metric proximal framework for a set of places denoted by $(X, \delta_{Phi}, d)$. Then consider a second metric proximal framework $(X, \delta_{Phi}, d)$. Let A be a subset in X. Then it is possible to define descriptive proximal mapping (follow Smirnov) f on A to Y. The mapping f is continuous, provided f(a) is descriptively adherent in f(A) in Y, if a is descriptively adherent in A.
Mathematics provides a means of structuring physical systems.
For example, if we consider a set of points X on the surface of an object, then X can be endowed with a proximity relation (e.g., Efremovic proximity relation). From this, it is then possible to derive a uniform topology on X. To do this, determine all subsets of X that are near each given set.
Again, for example, by endowing X with an equivalence relation, the resulting structure is a partition of X into disjoint subsets that are equivalence classes.
In general, mathematics provides a means of structuring nonempty sets that are independent of physical systems.
Dear @Issam, Mathematics, without exaggeration, does every thing. You need to find the proper approach.
Dear @Abedallah,
I am inclined to agree with you that Mathematics does everything as long as we can find the proper approach.
The London Mathematical Society has the following web page:
http://www.lms.ac.uk/research/what-does-mathematics-do
That page has lots of interesting links to other, related pages.
The London Mathematical Society has lots of interested related links:
See Mathematical Moments:
http://www.ams.org/samplings/mathmoments/browsemoments?cat=all
I consider Mathematics is the source of all sciences. Many domains depend on Mathematics.
It is well-known that mathematicians are discoverers of patterns and find various ways to structure space. An illustration of this is the following essay on how to construct Penrose tilings:
Penrose tilings tied up in ribbons:
http://www.ams.org/samplings/feature-column/fcarc-ribbons
An example of such a construction is creation of tiling by Penrose rhombs, with an intermediate result shown in the attached image.
Dear @James, thank you for the report about how to construct Penrose tilings; it is interesting.
Dear Abedallah,
There is a Mathematica CDF project that automates the construction of Penrose tilings with raised rhombi:
http://demonstrations.wolfram.com/PenroseTilingsAndWieringaRoofs/
The attached image(s) show several Penrose rhombi tilings using the CDF.
Dear James
Mathematics is indeed beautiful.
"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful."--J.H.Poincare
You have reminded me of another quote by G. H. Hardy:
"The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics."
Thank you James.
Dear Isaam,
What Poincare says about doing mathematics because one delights in it, hits the proverbial nail on the head. The beauty of patterns in mathematics captures the imagination and carries our thoughts forward to the extension of previous results and the introduction of new results. Both Poincare and Hardy have nicely summarized this penchant for and the magic in studying patterns in structures that we either find or introduce.
The August issue of the Notices of the American Mathematical Society includes an article to that may be of interest to followers of this thread:
H. Tamvakis, Mathematics is a quest for truth, Notices of the AMS 61(7), 2014, 703
One of Tamvarkis's salient points is that mathematics is characterized as a music of reason, not a cacophony of disconnected formulas. Another salient point is that mathematics teaches us to think critically.
The Septembre issue of the Notices of the American Mathematical Society contains
S.Mueller-Stach, What is a period?, 61(8), 2014, 898-899:
http://www.ams.org/notices/201408/rnoti-p898.pdf
This article illustrates in yet another interesting what mathematicians do, namely, classifying numbers in terms numbers rational or irrational or transcendental. Irrational numbers (e.g., pi, which cannot be written as a ratio of two integers) and transcendental numbers (i.e., such a number is not the root of any polynomial; e was proved to be transcendental by Hermite in 1873; pi was proved to be transcendental by Lindemann in 1882). There are many examples of transcendental numbers besides e and pi:
http://mathworld.wolfram.com/TranscendentalNumber.html
pi and e are examples of non-algebraic numbers because they are not solutions of polynomial equations with rational coefficients. Mueller-Stach carry this idea further by introducing what are known as periods. A period is a complex number whose real and imaginary parts are values of absolutely convergent integrals. It is still not known whether e or 1/pi are periods. The assumption is that they are not but that must be proved.
All mathematics and mathematics that are really useful and purposeful
@Issam Sinjab: Mathematics enabled us to 'see' an invisible process that is taking place millions of miles away deep inside the structure of these contact binary stars.
What does mathematics do?.
This discussion about What does mathematics do? is one of the richest on RG. It is the do-part of this discussion that I want to focus on in relation to work by topologists such as Paul Alexandroff (1920s) and Karol Borsuk (1940s), which has a direct bearing on the study of the configurations of the stars by Rene Descartes (1630s).
It was Paul Alexandroff who singled out the benefits of decomposing a surface into simple geometric structures call simplexes that are easily identified and measured. In the sciences, the focus is on local geometric planar regions and 3-dimensional regions of space. There are three Alexandroff simplexes to use as tools in decomposing a planar region of space, namely, 0-simplexes (vertices), 1-simplexes (line segments that can be either straight or curved) and 2-simplexes (filled triangles that can be either rectilinear with sides that are straight edges or curvililnear with sides that are curved).
Curvilinear filled triangles can either be convex (sides bulging outward) or concave (sides curved inward toward the barycenter of the triangles. The barycenter of a triangle is located at the intersection of the median lines (each median line is a straight edge between the midpoint of one side and the opposite vertex).
It was Karol Borsuk who called attention to the importance and practical benefit of decomposiing a geometric region into simplexes in the study of shapes. Borsuk observed that nerve complexes (also introduced by Alexandroff) are useful in measuring shapes, since nerve complexes tend to sit on shapes commonly found in visual scenes such as views of the stars. A nerve complex is a collection of simplexes that have nonempty interesection, e.g., collection of curvilinear triangles with a common vertex.
The decomposition of an region of space depends on our identifying vertices that can be triangulated. It was Rene Descartes (Le Monde, 1633) who first suggested who do this in his study of the stars. Descartes suggested that we select stars that can as vertices in a triangulation of the heavens. See, for example, the attached drawing by Descartes from his Le Monde:
https://www.jstor.org/stable/4025637?seq=1#page_scan_tab_contents
http://isites.harvard.edu/fs/docs/icb.topic1062138.files/Domski.pdf
I'm very surprised that one extended answer here is that a mathematician is looking for a "beauty" that is not easy to understand for me. I have open a question for trying to see what people can say about that
https://www.researchgate.net/post/What_is_beauty_in_mathematics_and_theoretical_physics
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