I respectfully disagree. Economics is nowhere near a branch of mathematics. They are two separate entities. Even as a physicist (which is one of the most math intensive disciplines), I don’t consider physics a branch of mathematics, and I’m sure most of my colleagues would whole heartedly agree. Matter of fact, many scientist see mathematics as a separate entity from science itself.
Scientist use mathematics as nothing more than a tool (a very powerful one at that). What distinguishes mathematics separately from physics is while physics employs a tremendous amount of mathematical rigor, it doesn’t use the scope or breathe of mathematics. For instance, in physics we deal with very complex equations called differential equations that contain complex solutions (answers). These answers may have variables or dynamics that a physicist may disregard simply because it doesn’t relate to anything physical or observable. The physicist is only worried about recreating reality through equation(s). Isn’t this what economist attempt to do? Mind you, economist does it awfully just like meteorologist or any other field encompassing chaos. The mathematician on the other hand, is interested in all aspects of the equation & doesn’t care about the constraints of the universe. The mathematician’s world exists in his/her mind that is not bound by the laws of nature. The mathematician considers all aspects of the math while the physicist is only concerned with what he/she needs to describe physical reality.
The most dangerous aspect about mathematics in the social sciences is that there is no underlying elemental structure to social behavior that serves as the basis for mathematical description. In physics and chemistry we have atoms, molecules, macro-aggregates, fields, internal symmetry, external symmetry, and so forth to form the basis for the mathematical construction of equations. This methodology works well because these equations have been formulated independently and the equations themselves have had the ability to predict with fierce accuracy. Or in other words, if there is an alien civilization out there, I’m pretty sure that they would have equivalent logic structures that would equate to the same mathematics and physics. Not so for social sciences.
Another aspect that has made mathematics so vital in physics and chemistry is the equivalency of the equations pertaining to various scopes of physical reality. No where is this more evident than with the Schrodinger Equation from quantum mechanics. Their exist three arenas to the study matter: Thermodynamics (the study of matter at the macro [bulk] scale-what me and you are most familiar with-think of trillions of atoms that make the world around you): Statistical Mechanics (the study of matter between the macro level and quantum level that takes into account large aggregates of matter but not at the level of thermodynamics-millions to hundreds of atoms): Quantum Mechanics (the study of matter at the atomic, molecular, and sub atomic level-a few atoms or less). The Schrodinger Equation, while not resembling the mathematics (formulation) of the equations in both statistical & thermo dynamics has an incredible property! When the Schrodinger Equation is taken too large limits, mathematicaly it transforms into the equations of statistical mechanics, and when those equations are taken to their limits- they translate to the laws of thermo dynamics and classical physics. I seriously doubt that economics has these underlying symmetries that are the central feature of the physical sciences. It’s these underlying symmetries that have made physics and chemistry powerful disciplines in unraveling the behavior of the universe.
Lastly, to develop equations of chaotic systems like the economy or weather systems deals with equations called complex partial differential equations or non linear dynamics. These equations are very difficult to find solutions too because they give a large array of answers for just a single input variable. This is the primary reason why climatologists aren’t able to predict the weather well. Complex systems are complex because they represent systems where population numbers are far too numerous, have too many variables, and too many degrees of freedom.
Any good scientist realizes the limitations of mathematics because it does have limits. Kurt Gödel (a famous mathematician) proved the limitations of mathematics by his famous Incompleteness Theorem, and since has given mathematicians and scientist a sense of humility and humbleness. I just wish some economist would learn Gödel’s theorem.
I must reiterate, just because a field is described by equations doesn’t mean its mathematics. Physics and Chemistry are described by very esoteric & complex equations that encompass many fields of math like complex analysis, real analysis, differential geometry, abstract algebra, group theory, Hilbert space, linear algebra… but in no way shape or form do we consider these disciplines to be a subset of mathematics. Economics doesn’t encompass the mathematical depth of the physical sciences… so how can economics be considered a sub set of mathematics? Mathematics is only a tool in describing phenomena unless it’s purely mathematics itself.
I apologize if any of the above doesn’t make sense. I’m in a hurry.
