Nerds are commonly arrogant in their youth. We pay attention to how things actually work and drive towards accuracy and truth/fact. In a sense it is the expression of dominance as a counter to the manners in which we are not dominant and this has been rubbed in our faces through a lack of mates, popularity, et cetera. However, it is an artifact of youth as a true nerd will quickly identify the limitations of his or her knowledge and thereby gain humility, will also gather a peer group in which they are not always the brightest and thereby gain the insight of having been a big fish in a small pond only en route to the ocean amongst the whales, and will experience the failures and manifested risks of arrogance learning to function in a world of people where our interests are joined and others have decisions.
Our strengths are long term in nature and, of course, not all of us successfully develop. We fail to continue growing and developing at our own peril. At the same time, the majority of our world's leaders and powerful people were once considered nerds.
As a reformed previous "brilliant jerk" one of the things I came to realize was that during that time my jerkishness was motivated by being requested and incented, even required, toward such behaviors but to be straightforward, it was also a result of my limitations of understanding and experience in working with people. What I needed was someone to help me understand the social side of the business and the greater value and advantage I could drive through greater diplomacy, empathy, and constructivity within my communications.
You might just find that you evolve the individual as well as the culture of the company while demonstrating loyalty and caring to and for your employees while retaining the talent and capability, accelerating broader productivity, and improving morale.
Mathematics is most basically a field encompassing the methods and concerns of quantification.
The field of computer science has largely been developed and expanded by mathematicians but I can only accept your assertion if you can demonstrate mathematics ability to address matters such as semantics or all of the other myriad concerns that are generally accepted aspects of the informatics field. Mathematics is a terribly handy and important tool in the toolbox but just one. The general companion idiom you may recognize is that if all you have is a hammer, the world is composed of nails.
My best formalization of how to express computer science is that it is "applied philosophy": the unification of the ideal with the concrete as a precursor of and mechanism towards the possible and actual manifestation of the unification of all ideals.
As a new product developer, I use math regularly. Most commonly I have used discrete, set, and stochastic maths. I am glad for the intellectual investments I've made in learning math.
Advice: if nothing else, you should ensure you are able to review and understand the academic presentations of solutions and methods or else you will have excellent solutions that you will not be able to take advantage of. Mathematical theories often describe the "shape" of problems and the factors involved with them as well as the challenges and limits you'll encounter. Without mathematics, you will reduce the top-end of what you can do and reduce your productivity at work. Few can identify it but you will cost massive amounts more for the businesses that hire you. On top of your greater cost, you will have to deal with more of the distracting and non-essential coding that composes the (in my opinion) more boring aspects of the field. Meanwhile, you will move more slowly than the rest of the market and loose any advantage you may have over time.
More personally, studying mathematics you will open up types of thinking and give yourself perspectives you might not otherwise have available. Even if your day to day doesn't require mathematics, you will be better positioned in life if you expend the effort to at least get a cursory understanding.
Keep all your developers focused on the problems you are solving. The really hard problems. And demand clarity in their code and solutions. In the end, the problem is the vehicle towards the value you are producing. Leadership's role is to define and communicate a vision of the value you produce, the engineer's role is to understand that vision before but necessarily thereafter identifying and solving the problems that expose their solutions and result in the delivery of the value.
If process or any other factor of the work place start to reduce developer focus from problem solving you will disable or lose your best people and accumulate slack-jaws. Having a job that actually keeps your brain revved up and challenged on an every day basis is crack to true geeks. Note that the basics like reasonable compensation, a good development machine, and basics like coffee et cetera are important to make sure no issue invades the mind space of your focused problem-solving employees.
Having worked in both a successful startup and a large corporation, this is what seems to hinge or turn any effort in our field.
All our basic learning functions are derived from emotional responses
The field seems rather convinced that it is a factor of the neural mechanisms that identify regularities and relationships in the environment. Likewise, it seems unprepared to harness those same methods for its purposes. Emotions are but one form of the perception of the environment, not unlike the "thoughts" or more direct perceptions (e.g. you see an object) we also experience. If I twist what I understand to be your meaning then I can state that we agree that the stimuli from the complete environment (including the representations of the environment that the brain provides) and your neural system's processing of it is in fact the basis of human learning. In a simpler and shorter statement: your absorption and comparison of the moments of your existence.
Otherwise, the notion that the abstraction of your sensory and extra-sensory experience is the basis of our learning is but a statement of your direct phenomenological experience.
We all started somewhere and frankly, if this drives some people to better contribute to themselves and the world or even just find the niche in life they've always wanted to be in, we'll have seen an excellent consequence.
A year of independent course work is unlikely to be enough to teach the automata theory/set theory/discrete mathematics/et cetera (ad infinitum) that is vital to developing a core understand of what one is interacting with in professional coding much less the various other "softer" disciplines required to know how to write code of a high level of quality. That said, even with many years of university, employment, and success behind me I am continuously learning, expanding, and refining myself. The risk, of course, is that low quality coders could result.
To counter-balance again, I generally like working on the harder and more interesting problems and this means that team members who are intimidated by those and as a result are happier with their job when doing the work that I do mostly because it needs to get done can be a godsend to my own happiness at work. Developing a mentoring relationship with such individuals has additionally been really gratifying.
Math is like love -- a simple idea but it can get complicated. -- R. Drabek