Was mathematics discovered or invented? Think deeply about it, did we create new mathematics or did we discover something that was already there, a deeper truth to the universe. Take for instance the Pythagorean theorem, where the squares of the two legs of a right triangle is equal to the square of the hypotenuse. One thing that is beyond doubt is that it was Pythagoras' genius that was able to see this truth that no one else did, and made an equation that modeled how it worked. In that aspect, the Pythagorean theorem was discovered, because it is a universal truth, which he discovered. Look at it this way... a^2 + b^2 = c^2 had never been written down so he did invent the formula, it was his creation. Pythagoras' theorem is the interpretation of a deeper truth, a human interpretation. Our nature as humans is to be imperfect, to err is human. 'a^2 + b^2 = c^2' is one humans interpretation of a deeper truth in nature. Although the person reading this may say they are an ashiest, as most scientists would say there are, I want you take into consideration an idea. Our creations as humans have all been imperfect models attempting to reach perfection which is ultimately unreachable. We as humans will always be imperfect, as that is our nature... the other side to our nature is growth in the direction of perfection, which could be interpreted as God. I promise I am not trying to be preachy, but the idea of God is the same idea as all knowing and perfection and it is also represented as that feeling inside us that drives us to be better people and make discoveries about the nature of the universe.
The question becomes more and more confusing the more you think about it. If we are creating things because we discovered them is that really a creation? Would someone be considered intelligent because they discovered an idea or because the developed a way to express that idea to others. I consider someone sufficiently intelligent if they are able to have complex ideas and explain them to someone who does not share the same intelligence. Let's look at some other genius mathematicians, specifically Joseph Fourier. Fourier was attempting to solve the problem of heat distribution, and through him solving the heat distribution problem it gave us a whole different way to approach problems. Fourier figured out that every function, continuous or not, could be represented as some combination of sine waves. Without his discovery we would not have anywhere near the signal processing capabilities as we do now, we would not have spectrograms or even the animation technologies we have today. One little idea can have such a large impact on a field, one little change in the way we think about something happening can exponentiate progress in a field.
When talking about artificial intelligence and deep learning, we are essentially trying to mimic human brain activity to give us human-like results. I will reiterate it again, our body and brain are machines that were perfected over billions of years of evolutionary modeling. A billion years is a very long time to develop a human, and we cannot expect to simply create a deep learning model to mimic its function. The least we can do is give it some help, some boost forward for its capability to process and understand information. The way to do this is through mathematics. Mathematics at its core is how things interrelate, it is the psychology of the universe, the explanation for everything. Mathematics is a truth of the highest order, and it is the most intelligent of us that are able to find it, communicate through, and explain these truths through the number and symbol system that we have developed. When it comes to some parts of math such as numbers in general, we did develop them, we created the base 10 system. Why do we count: "0, 1, 2, 3, 4, 5, 6, 8, 9" and then have the next number be '10'? We count with a base 10 system because we have ten fingers... there is no complex answer, it is just because we started counting with our fingers then we developed a more complex system off of this. Mathematics, the illustration of interrelation, exists beyond a base10 system; the genius of humanity is to connect with that fundamental truth and express it through our imperfect, human systems of language.
Things are easier to understand if they are in a more fundamental, readable form. We can extract more information more quickly from a spectrogram than we could from an air pressure versus time graph. And we can extract more information from a spectrogramic image that is generated to maximize both time precision and frequency precision than we would be able to extract from either a low or high bandwidth spectrogram. A spectrogram is really a genius innovation to the previous way of sound analyzation. A discrete Fourier transform is performed many times on a small window over the whole signal, and it outputs a three dimensional picture which we call a spectrogram. The third dimension of a spectrogram is normally represented with changes in color corresponding to changes in amplitude of the wave. The point is that something is easier to understand and extract information out of if it is presented in a way in which the fundamental elements are self-evident. Mathematics are essential for making these traditions, numbers and functions arranged in just the right way can transform chaos into order.
Because this is my blog and I can talk about whatever I want, I am going to talk about music. Beethoven, one of the most decorated, amazing composers of all time spent his whole career going deaf. There are patterns that are beneath and fundamental to the beautiful sounds. There is a fascinating relationship between music and math. Beethoven said, "I always have a picture in my mind when composing, and follow its lines". When we play a major chord on any instrument it sounds good, fulfilling almost. A major chord sounds good because of the matching frequencies of the notes within it. The bottom note completes 2 cycles in the time it takes the middle note to complete 2.5 cycles in the time it takes the top note to complete 3 whole cycles. This is also called a geometric series of waves. They all meet up at the same place, fairly frequently. Beethoven uses this technique called consonance, and it sounds naturally pleasant to our ears. Just as interesting to our ears is Beethoven's technique of dissonance. An example of dissonance would be notes a half step apart being played over each other. The sinusoidal shapes of both frequencies do not align with each other very well. The waves hardly ever, if they do ever, cancel out. Through contrasting consonance with dissonance, Beethoven adds that unquantifiable element of emotion and feeling into his music. Hector Barlios described one of Beethoven's finest works, "Moonlight Sonata 3" as "one of those poems human language does not know how to qualify". Beethoven's genius lied in his ability to feel the effect of the music and express it through music. One final quote for you to ponder on, by James Sylvester: "May not music be described as the mathematics of the sense, mathematics as music of the reason? The musician feels mathematics, the mathematician thinks music: music the dream, mathematics the working life."