Sacramento International Airport. Circa August 2017.
I flew solo into Sacramento International Airport. I was taking off right behind a Sky West... thrilling!
(NB: No safety was compromised when I took this picture)
I grew up on two computer games if you can call them so - Microsoft Flight Simulator, and Microsoft Train Simulator. The very idea of flying was thrilling and exciting for me. How does the world look from up there? How does it feel when the wheels kiss the runway (from the pilot's seat)?
On reaching Davis, I soon found out there was a small airport there, quite close to the main campus, closer to where I lived in my second year. Taking flying lessons was soon on my priority list, after academics, of course! This motivated me to look for a job. Luckily, I found one. Humble beginnings... a dishwasher at the coffee house at UC Davis - CoHo. I was soon able to find a position of a tutor, followed by that of a teaching assistant, and then an REU position funded by the NSF in about a year. Besides the occasional anxious call from mother, at odd hours, checking if I had crash landed, the urge to take flying lessons did a lot of good.
Life presents its teachers in most unexpected circumstances and non-obvious ways. After many bureaucratic problems and passing a host of background checks, I finally took up flying. This meant putting in overtime working hours during summer holidays and not going back home. I do not regret this because this hobby turned out to be one of the best teachers in my life. A pilot must trust the aircraft and yet take control when required. Moreover, flying requires hard work and unfailing dedication. Every flight is unique: variables like wind exist; pressures are never the same; no two aircraft have identical in-flight characteristics. I found this to be thrilling, and this is the reason for my continued interest in this hobby. I believe my interest in research is guided by similar reasons: it is dynamic; requires dedication and continued effort; provides room for personal and individual expression.
I am in India now. I miss flying already! I looked up on the internet and looks like there is, indeed, a flying club near the University of Cambridge. I will have to take a few examinations before I can fly solo or carry passengers in the European Air Space. Hopefully, I will be able to fly along with a test pilot. I am eager to fly amidst the English clouds.
Graduation Ceremenoy, University of California, Davis, CA. June 15, 2017.
My mum 'nd dad, cousin(she's a genius), and I; one of my friends, Yitong Li and his family.
Growing up, ‘intelligence’ was more of an if-and-only-if relation: intelligence if good grades. I took this to be true from the early years in school through most of middle school. However, this changed in high-school. To study topics taught in class and then to be able to reproduce that information in tests was apparently not the entire picture. While it has its own significance, a factory-line approach to learning and labeling intelligence in that way did not appeal to me. This motivated me to ask questions about the subjects I was studying which were often considered “not important for exams.” The internet became an excellent place to find answers to these questions, and I often found myself reading books which were not required texts for class.
I believe limitations posed by lack of resources are self-conceived. When one has a genuine interest in pursuing something and is prepared to accept the challenges on the way, these limitations cease to exist. Initiative and sincerity is the key. There is a definite trend of following traditional paths in whatever one does. The consequence of this is to accept conditions set by others. However, these formulaic approaches are surely not the only means. They play their role and are suitable for many. But I find it much more natural to not subscribe to these traditional paths. Every learner must explore and find their way. Towards the end of the second year at Davis, I felt limited by my major: Computer Science and Engineering. My interests were varied: abstract mathematics, statistics, and logic. But I did not have the option to take classes on these topics. So I decided to create my own major and Professor Aldo Antonelli agreed to be my advisor. To understand my interests better, I read many books and papers which were not directly needed for any particular class I was taking. I believe this was the best aspect of my undergraduate study.
Whilst reading on the history of mathematics and computer science I came across Hilbert, Gödel, and Turing. Their work appealed to me the most. I found what I was looking for: the confluence of computer science and mathematics. Indeed, the works of Turing and Gödel spring from within mathematics and are regarded as the firsts in the field of theoretical computer science. They were motivated by philosophical reasons and formalized within a mathematical framework. Although I was unable to declare my own major, I decided to study two majors: Computer Science and Philosophy.
I graduated a few days ago and was awarded a... "Bachelor of Arts and Science, Highest Honors in Computer Science, Highest Honors in Philosophy." It is a mouthful. I had to train myself to say the entire thing in one breath! I was also awarded the Chancellor's Award for Excellence in Research. The best thing that happened as a consequence: my parents had reserved seating at the convocation ceremony. I was so relieved and delighted to learn that they could not have to queue to secure seats in the front row.
TIME published the list of 100 most influential people of the century on January 1, 2000. Only two mathematicians were featured on the list - Alan Mathison Turing and Kurt Godel. Indeed, what these men had done would shape the time to come. Turing's work (Turing, 1936) which had greatly inspired von Neumann, after whom the John von Neumann architecture is named, proposed the first RAM model - the remnants of which are still found in modern computers. That said, I have recently read scores of papers on Turing's thesis besides the second-hand books on Turing I found at a book shop. Two words with almost the same spelling has interested my: computer, and computor.
Every word must be treated with respect and used decorously. Here, I am interested in two words, which are in very close proximity to each other. They are - "computer" and "computor". The word "digital-computer" is often used to emphasize the difference between them.
Alan M. Turing's "computer" and some "computer" on which you are reading this are different. In the English Oxford dictionary, the first meaning of the word "computer" is "An electronic device which is capable of receiving information (data) in a particular form and of performing a sequence of operations in accordance with a predetermined but variable set of procedural instructions (program) to produce a result in the form of information or signals." Then, there is the other meaning which is often overlooked - "A person who makes calculations, especially with a calculating machine." On looking further down, you will find the word "computor" with the word "history" flanked in parenthesis, demarcating that it is an archaic word, or perhaps obsolete. This is what the word means - "A person who makes calculations or computations". To add to the confusion, it also means "A machine for performing computations". Note that the word "computor" is also quite close to the word "computator" which has a similar meaning.
Recall that we have two words under consideration now. They are "computor" and "computer". Now, I will introduce another word. The last word on the list I had presented at the very onset - "digital computer". The word "digital" means "(of signals or data) expressed as series of the digits 0 and 1, typically represented by values of a physical quantity such as voltage or magnetic polarization." It followed by "computer", then means, "A computer which operates on data in digital form; contrasted with an analogue computer."Turing's papers are unique in the way they are written - often informal and conversational. Here's (Turing, 1950 ) one of his great papers where he cosiders the question, "Can machines think?"
In the first half of the twentieth century, two distinct views of understanding the role of axioms in mathematical models and proving their consistency and independence results evolved. The first view was the one held by Gottlob Frege and the latter by David Hilbert. Although Frege and Hilbert, both believed that mathematical theories should be axiomatized and that any reasoning in mathematics should be devoid of vague intuition and thoughts, their views had fundamental differences. Mathematical reasoning must be clear and rigorous and follow certain rules and every step must be unambiguous and explicit. However, Hilbert “. . . believed that Frege’s attempt to build mathematics on self-evident foundations was misguided.” Hilbert’s main concern was to prove the consistency of his theory and show that it was free of contradictions. Hilbert, in his Grundlagen der Geometrie, lays out the precise set of axioms for Euclidean geometry. Hilbert’s system, H, consists of three primitive terms – i) point, ii) line, iii) plane and three primitive relations – i) Betweenness, ii) Containment iii) Congruence. H has 21 axioms – categorized under Incidence, Order, Congruence, Parallels, and Continuity. Hilbert demonstrated several results on the independence of the axioms in the theory and proved some fundamental theorems of geometry. However, Frege had a very different view. For Frege, axioms are true statements. According to him, axioms must be self-evident and must be true. Since Hilbert’s axioms had the primitives (which were undefined) in them, they cannot have truth values and hence were unacceptable to Frege. The axioms were open to interpretation and hence do not express proper Fregean thoughts. Frege’s logic is, and so unlike Hilbert, he does not have to worry about consistency.
Contrary to Immanuel Kant, who regarded mathematical (arithmetical and geometrical) propositions as examples for synthetic a priori propositions, that is, propositions that are not empirical, but enlarging knowledge, Frege wanted to prove that arithmetic could completely be founded on logic, that is, that each arithmetical concept, in particular, the concept of number, could be derived from logical concepts. Arithmetic was, thus, analytical. Frege’s idea behind the programme was to provide an objective and concise mathematical framework. He intended to protect Arithmetic from Psychologism (German: ’Psychologismus’) and Formalism.
Frege emphasized the need for a formal system, free of contextual interpretation, to carry out mathematical reasoning. In 1902, Bertrand Russell found Frege’s mathematical system to be inconsistent. However, it cannot be denied that the Logicist programme was unique and farsighted. Initially, Frege’s works did not get much recognition mainly due to his two-dimensional notations. However, the formalization of mathematical reasoning and the explanation of logical concepts in his works were essential to the development of ”Principia Mathematica“ by Russell and Alfred North Whitehead, and Kurt Gödel’s incompleteness theorems. This later paved its way in 1936 when Alan M. Turing (now regarded as the father of computer science), using his concept of a Logical Computing Machines (now called Turing Machines) provided a negative solution to the Decision Problem posed by Hilbert in the early twentieth century. Frege’s system of Logic had a profound impact on the future developments of formal systems in the twentieth century and later.