Ever since the financial crisis of 2008 there has been some discussion about whether or not economics is a science. Some, like Russ Roberts of Econtalk, channelling Friedrich Hayek, do not believe that economics is a science. They think it’s more like history where we come up with descriptive narratives that cannot be proven. I think that one thing that could clarify this debate is to separate the goal of a field from its practice. A field could be a science although its practice is not scientific.
To me what defines a science is whether or not it strives to ask questions that have unambiguous answers. In that sense, most of economics is a science. We may never know what caused the financial crisis of 2008 but that is still a scientific question. Now, it is quite plausible that the crisis of 2008 had no particular cause just like there is no particular cause for a winter storm. It could have been just the result of a collection of random events but knowing that would be extremely useful. In this sense, parts of history can also be considered to be a science. I do agree that the practice of economics and history are not always scientific and can never be as scientific as a field like physics because controlled experiments usually cannot be performed. We will likely never find the answer for what caused World War I but there certainly was a set of conditions and events that led to it.
There are parts of economics that are clearly not science such as what constitutes a fair system. Likewise in history, questions regarding who was the best president or military mind are certainly not science. Like art and ethics these questions depend on value systems. I would also stress that a big part of science is figuring out what questions can be asked. If it is true that recessions are random like winter storms then the question of when the next crisis will hit does not have an answer. There may be a short time window for some predictability but no chance of a long range forecast. However, we could possibly find some necessary conditions for recessions just like cold weather is necessary for a snow storm.
Scientific Clearing House 
Mathematician Michael Edesess has an interesting essay in which he says economics uses definitions that are too imprecise to allow for useful conclusions to be drawn.
You, as a mathematician, may enjoy reading his essays.
http://ineteconomics.org/blog/inet/economics-not-math
The other interesting thing that loosely pertains to your post is this entry by economist Stephen Keen. About half way into his post, he summarizes an interesting paper entitled Motivated Numeracy and Enlightened Self-Government. This paper found that people with higher numeracy skills could interpret a given set of data if it applied to a mundane topic. Yet, those with the highest numeracy skills surprisingly misinterpreted the exact same data if it was presented in a politically important context. In other words, a person’s higher math skills did not protect them from politically biased interpretation of data.
http://www.businessspectator.com.au/article/2013/12/9/economy/when-economic-theory-fails-maths-exam
This is fascinating to me. Unfortunately, economics and politics are frequently mixed.
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according to an article from 2010 (12 (6) in the e-journal entropy (open acces) ‘fairness is an emergent self-organized property of the market for free labor’ (he doesn’t actually do some of the math details, but refers you to somebody), which is actually a very old result—and i called him on it (he’s at purdue, perhaps a competent person with some skills, who’s also a cranck )—and its wrong too.
you can also look at ‘paul cockshott’ ‘conservation laws, financial entropy, and the eruozone crisis). i’ve interacted with him too (he had some conference in the uk—a place that was discovered by native americans arrund 1492—with machover (israel—‘laws of chaos’, pretty wordy and basic) and duncan foley (NSSR)—exactly the same if not worse, has some papers with e smith of sfi) . i asked about solutions for the fokker-planck equation (updated of course, such as donald ludwig ubc 1977 SIAM review, the ‘thermodynamic hamiltonian’), and he said he’d never heard of it, and said i should study some ‘garbage’ computer science language.
i referred him to a certain equation in a book or paper with his name on it, which was the fpe. he’s a communist too!!!
‘from each according to all, to each according to (name your hood)’.
you can also look at ‘cliodynamics (peter turchin, pnas 2013) and clioemtrics, not to mention jurisdynamics.
some guy who writes sometiimes in NYT mostly on sports but has good computer data skills, proves your point—he beat ray fair (yale, fairmodel) on wikipeida predeicting the election.
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@Tom I can’t say I agree with Edesess. The simple models he attacks are what would be called “effective theories” in physics and can be extremely useful. I use them all the time when modeling biological systems. He is correct that these aggregate quantities are difficult to measure but these Keynes-Hicks type models can be very useful in getting a qualitative understanding of the situation. Thus far, the conclusions about what happens at the interest rate zero lower bound have been validated. Quantitative easing is not leading to runaway inflation as predicted.
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