Yes. Pure mathematics is useful for theoretical physics. Mathematics is nothing but  logical expression of physics and physical things. It is much more clear than physical logic if the methodology is right. A physical concept has to be converted to mathematical equation and the mathematics will drag it to a final equation which can explain a new concept which is not visible to the conceptual logic. Conceptual logic must provide a physical phenomena which can be observed and verified physically. Otherwise that mathematics will create complicated and chaotic outputs in theoretical physics..

Yes. Pure mathematics is useful for theoretical physics. Mathematics is nothing but  logical expression of physics and physical things. It is much more clear than physical logic if the methodology is right. A physical concept has to be converted to mathematical equation and the mathematics will drag it to a final equation which can explain a new concept which is not visible to the conceptual logic. Conceptual logic must provide a physical phenomena which can be observed and verified physically. Otherwise that mathematics will create complicated and chaotic outputs in theoretical physics..

I agree with the answer given by Dr. Siva Prasad. Physics is based on observations, but quantifying physics needs logical methods, and mathematics just provides that connection. It may start as pure mathematics, but it can find applications when certain phenomena are to be modelled. 

Hello,

I agree with this assumption. Theories like quantum physics or cosmology (black holes concept, universe expansion) are fully abstract and beyond our senses possibilities. So maths in a way bring ideas and tools to solve physics questions and in another way are a training to think the physical concepts.

Gabriel

Mathematic is definitely required. Mathematics is only a language in fact that can describe anything we can imagine. With respect to theoretical physics, the limits are set by experimental observations.

For example, from Newton's equation, it may appear that velocity of a body can possibly increase indefinitely (mathematically described theory). But we know now that this is impossible. So the equations were modified to account for the now known limit (mathematically described theory fine tuned to match a physical fact).

I also must say that the mathematics is absolutely required. But the theoretical physics do not grow via mathematics. It do not enter a new paradigm via mathematics. It goes forward by inventing new concepts and begining to look at Nature differently. Marhematics acts next as a logical language and almost a tool. 

Here are Wigner's thoughts on this: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

I've always liked his conclusion: "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."

You should maybe also have a look at https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences

Not all mathematics represents physical, experimental reality. How does one separate the mathematics that represents physical reality from that which does not represent physical reality? The answer of course is to preform physical experiments that end in measurements of a physical variable.  Strictly speaking pure mathematics is based on axioms which strictly speaking do not represent physical, experimentally determinable reality.  In order to represent physical reality the mathematics must be based on experimentally determinable postulates not axioms.  In my own work in theoretic physics (www.jmkingsleyiii.info) I use functional analysis and the calculus, both of which are based on axioms not postulates.  However both can be place on a postulational basis starting with substituting small spheres of radius e>0 as small as necessary for the experiment at hand but not e=0 to represent points and right circular cylinders of radius e>0 as small as necessary for the experiment at hand but not e=0 to represent lines and so on for planes and the other constructs of the calculus and functional analysis. 

The idea of using axioms instead of postulates as the basis of pure mathematics goes back some 2400 years to Euclid.  Of course plane geometry is useful but strictly speaking it does not represent physical reality.  In order to represent physical reality, one should start by replacing points, lines and planes by their 3 dimensional analogues as above. 

The problem is that some pure mathematicians have forgotten that in order to represent experimental physical reality, the mathematics and the mathematical results must yield real numbers that are equal to experimental results.  The result is that some 2400 years after Euclid, some string theory mathematicians consider time to be an imaginary number but they have no idea as to how to measure imaginary time and thus for them, time is simply a dream with no physical correlate.

J.M. Kingsley III

The scientific method has hampered physics more than it served. Physics needs mathematics in order to explain things. Experiments never explain. At the utmost they describe. Models are needed in order to plan the setup of the experiments and in order to interpret their results. These models are mathematical models. The foundation of physical reality is not accessible to observations and that includes measurements. This foundation can only be investigated via mathematical test models.

Yes the language and logic of mathematics resolves the originating conscious idea that then leads to some truth of reality.

Try: http://www.e-physics.eu/MechanismsThatKeepRealityCoherent.pdf

But not all mathematics represents physical reality and the use of experiments is absolutely necessary to separate that mathematics which represents physical reality from that mathematics which does not represent physical reality.

Not all mathematics represents reality. The trick is to find which part does represent reality. Another fact is that mathematics is not complete. For example multidimensional integration is far from mature. The darkest part of physics are mechanisms that ensure the dynamical coherence  of physical reality. Physics neglects these mechanisms, but without such mechanisms universe would quickly turn into complete chaos.