Try The Mathematical Tripos

The Cambridge Math Tripos has existed in some form since the 1700s and the students used to be publicly ranked by exam results so it's basically a 300+ year old competition. We have full access to detailed syllabi, archived notes, example sheets (assignments), computational assignments, and exams. Any books needed use archive.org, a local library, Anna's Archive or Library Genesis though many of the books we need will be freely available on the author's site.

You can do this by hand and ask to have your worked checked on math stackexchange or you can try what I'm going to do which is an insane attempt to prove it all using the Lean 4 proof assistant and whatever other proof assistants like Agda, Coq, Isabelle, even try the new experimental proof assistants replacing sets with types in Part III if we ever make it that far. I'm also going to build a proof assistant in the first course Numbers and Sets if you're interested but of course you don't have to do any of this and can skip it continuing on to Tripos glory.

For geometry themed entertainment try Isaac Newton on Mathematical Certainty and Method by Niccolo Guicciardini.

Isaac Newton entered Cambridge Trinity College in 1661 for it seems a medical degree as a sub-sizar meaning subsidized tuition in exchange for being a domestic servant on campus waiting tables and working in the kitchen, whatever labor that was needed. Cambridge then was just a diploma mill there were seldom any lectures, the oral exams at the end of your degree basically optional, and fellows (graduate students) took jobs as tutors to supplement their income when they weren't getting drunk in nearby taverns. Undergrad degrees consisted of one of these tutors giving you a list of books to read. According to his Trinity College notebook which Cambridge still has this curriculum was a large reading list of classics such as Du Val's four volume collected works of Aristotle like Organon and Metaphysics, Porphyry's Isagoge, Vossius's Rhetoric, Diogenes Laertius, Epicurus, Plato, and many Roman poets like Ovid. This was because of the recovery of Aristotle in the middle ages which shaped 'philosophy of natural sciences' courses for the next few centuries. There didn't exist titles like mathematician, practitioners then considered themselves philosophers. Newton soon stopped taking notes of his official curriculum and must have read Walter Charleton’s Physiologia because he wrote down 37 headings on different pages that were questions to be answered investigating natural science topics, 18 of which came from Charleton's book and some from Aristotle. These were 'Of Atoms, Of a Vacuum, Of Vision' and many others. He gave each topic different sized gaps indicating how much he thought he needed to write about each. Everytime he would learn something about them he'd write notes under the heading like a comet position he once tracked.

According to Newton's notebook and the writings of his close friend De Moivre here is how Newton learned mathematics. Wandering the town fair in 1663 he comes across a book on astrology and out of curiosity of the occult buys a copy. There's a figure he couldn't understand in the book because he didn't know trigonometry. He buys a book on trigonometry but couldn't understand the demonstrations because of a lack of geometry. At Cambridge he goes to the math department and everyone there is deeply immersed in the work of La Geometrie by Descartes. They tell him to buy Barrow's Euclid Elementorum an 'easier' or explained version of Euclid's Elements. He skims it and after finding what he needs for the astrology/trig book abandons the book as 'too trivial demonstrations' (later has to go back and relearn classic geometry).

He returns to the math department and is given Oughtred's Clavis mathematicae or the key to mathematics writing he understands it except for the solutions of quadratic and cubic equations however his takeaway in his notes is that algebra can be used for exploration, which he starts doing by writing out hundreds of examples.

Again he returns to the math department to ask for a new book, and they are all still immersed in Cartesian geometry so Newton, despite being told it was a very difficult book, borrows a 1659 two volume Latin translation of Descartes La Geometrie by van Schooten called Geometria, a Renato des Cartes with appendices and commentary by his students. This book was considered the state of the art of 17th century analytical geometry, it would be like reading a graduate text today by a leading expert that had an appendix full of lecture notes and PhD student dissertations.

This is the algorithm Newton used to read van Schooten's two volume book. He read a few pages or so, couldn't understand the text, and went back to the beginning. Went a little further then stopped again going back to the beginning. He repeats this loop by himself until he finally 'makes himself whole of Descartes'. Looking at the online copy of this 1659 version the Descartes geometry is about 104 pages and 450 or so pages of additional research/commentary. It is broken up into 3 parts and assuming he didn't have to reread each part everytime he looped, only the relevant part (~34 pages), it probably took him 3 months to finish van Schooten which is what his notebook shows that after a few months he was already doing research in analytical geometry trying to generalize Descartes. His notebook indicates he learned from the grad student commentaries how to transform a hard problem into a different simpler one. Somewhere around this time he moves dorms complaining his peers were too busy partying so lives with more serious students.

Newton's notes which you can read online show him trying to generalize any math he read which often would lead him into algebraic corners where he would get stuck as there didn't exist at the time methods to solve/simplify. He seemed to assume everything he read was a special case of something more general such as the insight that integration is the reverse operation of differentiation so there must only be a theory of derivatives somewhere waiting to be discovered.

After Descartes he returned the borrowed copy and bought a different copy of Descartes for himself, and van Schooten's Exercitationes mathematicae libri quinque or Five Books of Mathematical Exercises to help him fill in the blanks of his algebra misunderstandings as he had a lot of mistakes in his notebooks regarding negative roots. Newton assumed the cubic parabola was the same shape in all quadrants but soon after corrected these mistakes. Descartes geometry wasn't like today's Cartesian plane it originally consisted of just one positive quadrant but van Schooten and his students had expanded it.

Descartes' La Geometrie is filled with little comments of encouragement as he was afraid nobody would read his work if it was too long so everywhere there is these reassurences like 'don't worry reader the following isn't too hard' or insisting the reader figure out a problem themselves to not deny them of the satisfaction he found figuring it out himself. The different copy Newton buys has all these comments left in tact and Newton misunderstands, or the translation is incorrect, one of these comments by Descartes about the equation of curves stating 'it is easy' to find everything you want to know about a curve from it's equation and the reader need not be bothered by a lengthy demonstration when in fact this is an unsolved problem. Newton can't figure out this supposedly simple exercise in Descartes' book so breaks down the equation of a curve into many headings in his notebook and tries to generalize which led to him creating his own advanced analysis well beyond any other mathematician at the time.

Around the same time when he was trying to figure out that Descartes exercise he reads Arithmetica Infinitorum by Wallis which is the arithmetic of infintesimals. Newton recognizes many of the sums are similar to what he read in Oughtred's book, known today as Pascal's triangle and in typical fashion seeks to generalize and invents the binomial theorem. Newton then reads Viete's Opera Mathematica which was another textbook by van Schooten compiling all the work of Viete such as Diophantine equations. In less than a full year Newton managed to bring himself up to date with the entire achievement of mid 17th century mathematics by himself and begins self-directed research writing out 22 headlines of 'problems' in his notebook and classifying them into groups regarding integration, analytic geometry and mechanics.

In April 1664 Newton has been a subsizar for three years and has to apply for a new scholarship at Trinity College where he first meets the Lucasian Professor Isaac Barrow who examines him on Euclid and finds he knows nothing about geometry. However Barrow had also worked on infinitesimals and apparently had invented some of calculus himself but not noticed if you read the book The Geometrical Lectures of Isaac Barrow. The author found in Barrow's notes he definitely had a kind of proto-calculus worked out but didn't seem to notice or care about it's analytical importance enough to write a treatise about it. Through these conversations no doubt Barrow recognized the potential of Newton and unlike modern day academia where he would have tried to squash Newton and sabotage his work he becomes Newton's mentor and starts giving him work to do like helping to publish Barrow's books and giving Newton all his research on infinitesimals. Barrow also helps Newton obtain his scholarship in exchange for the agreement Newton attend his lectures on geometry and learn the importance of Euclid.

The plague shuts down the school from the summer of 1665 to spring of 1667 and Newton returns home, makes himself an office by building bookshelves for his now large library and spends all his time doing research with his new analytical tools building them into modern calculus. Describing his activities during the second plague year: "I am ashamed to tell to how many places I carried these computations, having no other business at the time, for then I took really too much delight in these inventions". There is notes he kept of calculating a logarithm to it's 52nd decimal point. This is basically the end of story for Newton's analysis research, sometime during the pandemic years he is satisfied with his calculus and abandons research in math to pursue his other questions in his notebook. There is a lot of notes about how tedious it was to do calculations before he came up with his analysis so we can assume he became satisfied that he had all he needed.

Newton believed the ancients had already figured everything out, and this information was lost over time after disasters had destroyed the information. He based this from his own experience living during the plague, the great fire of London and political upheaval. This led him to alchemy where he believed their knowledge was encoded in myths written in Roman and Greek literature where every myth was real but that their lives were embellished through story telling. A common alchemist practice for example was to interpret Ovid's Metamorphoses where the god of the forge/metalwork Vulcan catches his wife Venus and Mars locked in an embrace so traps them in a fine metallic net. The alchemists of the Royal Society that Newton belonged to frequently used the names of planets for metals so obtaining an alchemists manuscript by George Starkey he recreated this myth and ended up with an alloy with a strided net like surface. Newton also decoded Cadmus and the founding of Thebes from Ovid into practical lab instructions.

20 years later Newton has produced numerous manuscripts but refuses to release any of them until he is offered to write the Principia, and wanting to prove infintesimal calculus he tries uncovering the ancient analysis used in classical geometry. Pappus' book 7 contains a commentary about the tools and propositions that Euclid, Apollonius, Eratosthenes and other geometers of the day used that Pappus referred to as the 'treasury of analysis'. These were said to be contained in books by Euclid but those were never recovered. These lost Porisms (corollaries) are speculated by Newton to be projective geometry.

Newton in his Lucasian chair lectures said that the ancients would never bother to introduce the algebra of curves with geometry because you lose the simplicity of working within geometry as its whole point was to escape the tediousness of calculations by simply drawing lines and circles. He also claims books like Pappus's Collectio deliberately hid the analysis, as it was considered an inelegant tool and that ancient synthesis where they deduced a consequence from a given premise (a corollary) using geometric demonstrations was a superior method as the analysis could not be reversed in steps like geometric synthesis could. He went further and claimed if you wanted to discover seemingly unrelated corollaries you had to use synthesis, describing the analysis of his day as a 'tedious pile of probabilities used by bunglers'. In other words if he didn't rely on geometry he could not have found most of the critical results of the Principia. People today claim he did this to avoid priority disputes over his analysis he kept hidden except for privately circulated manuscripts, but he used geometry to invent limits and plane transformations that were not formally developed until 200 years later so he was probably right. His limits are very similar to what everyone today is using in modern calculus courses. If you take a calculus course today they will tell you Newton's influence to promote geometry allowed continental mathematicians to freely develop analysis which kept back Britain decades in the field of analysis.

He later wrote in a manuscript on geometry that mechanics of motion was what generated all geometry and that the ancients had understood this as well conceiving geometrical objects as generated by moving along a straight edge, circles/elipses via the movement of a compass, or via translation like in Proposition 4 of Book 1 of Euclid's Elements where one triangle form is moved to compare to another. This is where he demonstrated that rotation of rulers were in fact transformation of the plane. Here is how Newton used his rotating ruler to create a power series.

"Comparing today the texts of Newton with the comments of his successors, it is striking how Newton’s original presentation is more modern, more understandable and richer in ideas than the translation due to commentators of his geometrical ideas into the formal language of the calculus of Leibnitz" -Vladimir Arnold 1990

Newton's completely different life then began at age 52 when he left Cambridge and went on to become both the warden and master of the London mint for 30 years a highly lucrative job with enormous bonuses. He cracked down on mint corruption and absenteeism by personally overseeing the wagies and making them perform tedious tests on batches while producing highly pure coinage due to his dabbling in alchemy. Newton had a Professor Moriarty tier nemesis. Due to his new standing in London elite circles he's made president of the Royal Society and was hilariously ruthless ruling it with an iron fist vanquishing all his previous critics. His niece came to live with him in London and was some kind of famous socialite of the time holding many parties at his house where he'd entertain guests with his expensive cider that he had made buying apple trees from Ralph Austen of Oxford a renowned cider maker. Newton was famous for wit and performing dramatic demonstrations which Princess Caroline enjoyed so much he was a regular feature of her court. He was not the miserable loner that modern day biographers wish to portray him as. His letters show regular payments and gifts to all extended family members too so he wasn't a scrooge and he didn't die broke either from an investment scam. His estate was worth £30,000 when he died and to see how much that is worth today the average yearly salary of a Surveyor in London then was about £131 so he could have employed 229 Surveyors for a single year. Today an average salary for a Surveyor is ~£60k per year and employing 229 for a year is almost £14 million.

According to a Cambridge historian who wrote Life after Gravity Newton gave as a dowry to his niece and her future daughter a 200 acre estate which he later moved into when he was dying. He had an elaborate sun dial built in the gardens:

A very curious relic of Sir Isaac survives in the garden at Cranbury Park, viz. a sun-dial, said to have been calculated by Newton. It is in bronze, in excellent preservation, and the gnomon so perforated as to form the cypher I. C. seen either way. The dial is divided into nine circles, the outermost divided into minutes, next, the hours, then a circle marked "Watch slow, Watch fast," another with the names of places shown when the hour coincides with our noonday, such as Samarcand and Aleppo, etc., all round the world. Nearer the centre are degrees, then the months divided into days. There is a circle marked with the points and divisions of the compass, and within, a diagram of the compass, the points alternately plain and embossed.

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• YouTube - Arithmetic and Geometry Math Foundations 2

The successor function s(n) = n + 1 for natural numbers:

• s(0) = 1
• s(s(0)) = 2
• s(s(s(0))) = 3

Let's prove the laws of multiplication using his definitions

Prove n * 1 = n

The definition of multiplication: mn is n + n repeated m times so n1 is 1 + 1 repeated n times so 1₁ + 1₂ + … + 1_n = n

We will need this result for the other proofs.

Prove the distributive law: (k + n)m = km + nm

km is m + m repeated k times or m₁ + … + m_k and (k + n)m means m + m repeated (k + n) times. Distributing m in (k + n)m we get: ((m₁ + m +…m_k) + (m₁ + m +…m_n)) and now the left side matches the right side since km + nm is the same when expanded.

Prove the associative law: (kn)m = k(nm)

(kn)m is m + m repeated kn times and kn is n + n repeated k times:

(n₁ + … + n_k)m or (mn₁ + … + mn_k) using distributive law.

Right hand side: k(nm) is nm + nm repeated k times (nm₁ + … + nm_k) and factor out m, both sides of the equation are the same since we haven't proven the commutative law yet so can't claim nm sequence is the same as mn sequence.

Prove the commutative law: nm = mn

First let's prove n1 = 1n or 1n = n. We proved n1 = n already, so we can use substitution to replace n in 1n with 1(1₁ + .. + 1_n) and using the distributive law this is (1₁ + .. + 1_n) or n. Now we have n1 = 1n.

Looking at the right side: mn is (n₁ + … + n_m)

Factor out n: n(1₁ + .. + 1_m) and a sum of 1's up to m is m, we have nm = nm.

• YouTube - Arithmetic and Geometry Math Foundations 4

Try the subtraction laws

• n - (m + k) = (n - m) - k
  • The left side is n + (-1)(m + k)
  • (-1)(m + k) via distributive law is (-1)m + (-1)k
  • (-1)m is m + m groups at -1 times which makes no sense so we can commute
  • m(-1) now it's m copies of -1
  • n + (-1₁ + -1₂ .. + -1_m) is -1 added up m times or -m
  • n - m + (-1₁ + -1₂ .. + -1_k) is -1 added up k times or -k
    • n - m - k = n - m - k

• YouTube - Arithmetic and Geometry Math Foundations 6

A simplified Roman numeral system is introduced

• Uses powers of 10
  • 327 = 3 x 10² + 2 x 10¹ + 7 x 10⁰

• YouTube - Arithmetic and Geometry Math Foundations 7

A new way of hand multiplication is shown, taking advantage of the fact the notation is using the distributive law. "Borrowing" while doing hand subtraction now makes sense. Wildberger rants how his daughter was taught a ridiculous way in the questionable Australian modern school system.

• YouTube - Arithmetic and Geometry Math Foundations 8

In every calculus course like the recorded lectures for MIT's 18.01 the prof repeatedly uses these laws without explaining them manipulating equations on the board. Wildberger says he didn't find any long division in his kid's curriculum because schools removed it claiming it was 'too difficult'. Another way to do long division:

7)203

How many groups of 7 x 10² fit into 203? 0 so reduce to how many groups of 7 x 10¹ fit into 203? 20

   20  
7)203
 -140
   63  

How many groups of 7 x 10¹ fit into 63? 0 so reduce to how many groups of 7 x 10⁰ fit into 63? 9

   29
7)203 
 -140
   63
  -63

• YouTube - Arithmetic and Geometry Math Foundations 9

Interesting how he defines all these as types.

• YouTube - Arithmetic and Geometry Math Foundations 10

The prime notation used in the proof represents another similar variable.

• YouTube - Arithmetic and Geometry Math Foundations 11

More simple proofs.

• YouTube - Arithmetic and Geometry Math Foundations 12

The point in watching these is to see how you can start with just the natural natural numbers and keep going defining more and more complexity with new types of numbers like integers.

• YouTube - Arithmetic and Geometry Math Foundations 13

We learn what a field is. If you're curious in the introduction here he shows how the rational number line has been raised on the y-axis showing how all numbers on a line through the origin (0,0) are equal on the rational strip. It explains why a/1 is a and why a/0 is not defined.

If for whatever reasons you want even more arithmetic try Arithmetic by Herb Gross. This is vocational school style math that was originally taught to adults at community colleges and in prisons in North Carolina. The book is here archived and the YouTube playlist has exercise explanation videos you don't have to watch unless needed but they're very good he teaches everything using bounds and inequalities or relating concepts to English grammar. A free resource for practice is expii it was designed by the US olympiad national coach Poh-Shen Loh. We will do so many problems however in the tripos that you will pick this all up anyway.

Logs were invented to simplify large multiplications for example spherical trigonometry (navigation, astronomy). Data visualization will typically scale the axis with logs in order to fit the data into a presentation without needing gigantic numbers on the axis. Logarithms are linearized exponents and all the rules of exponents are the same for logarithms such as: \(\frac{c^a}{c^b} = c^{a - b}\) or log(a/b) = log(a) - log(b).

Log laws:

\[\log(ab)=\log(a)+\log(b)\] \[\log(a^b)=b\cdot\log(a)\]

These can both be combined:

\[\log\left(\frac{a}{b}\right)=\log(ab^{-1})=\log(a)+\log(b^{-1})=\log(a)-1\cdot\log(b)\]

A base 2 logarithm sometimes written lg(x) can represent how many times do you divide by 2 until you reach 1 or less. A base 10 log can tell you how many zeros are in a number like log(1million) is ~6.

A base e log or ln(x) is often used instead of base 2 or lg(x). This is due to \(log_{2}(x) = \frac{ln(x)}{ln(2)}\) meaning you are off only by a constant factor and now can enjoy all the nice properties e^x and ln(x) have for doing quick approximations by hand like the derivative of e^x is itself and the derivative of ln(x) is 1/x. For small inputs where x² is smaller than x then e^x can be approximated with 1 + x. To see how e^time = growth and ln(growth) = time then read this which explains why lg(1) = 0 since if you have 1x growth then time hasn't started and e⁰ = 1 or 1x growth at time 0.

The Tripos consists of Part IA, IB and II and is supposed to take 3 years to achieve a BA honors degree. Part III is a one year masters degree similar to a first year PhD student taking 5-7 grad courses. It is a fully immersive degree where all you do is math 6 days a week there is no other electives. Each year has 3 terms that are 80 days long: Michaelmas, Lent and Easter. Each term is 4 courses of 24 lectures each except Easter where computational projects are done and the exams. This is why it's considered a very hard degree because imagine taking 8 courses and having to know them so well that you can pass a series of 3 hour tests many months later.

During the term you work through 4 example sheets per course which is how we'll know that we are keeping on track. These are usually due every 6 lectures and attending lectures is optional some just read the notes and work through the recommended books. Each course has around 4-6 recommended books usually the first 2 represent the first half, the others the second half. There's no TAs, no recitations, no midterms, no office hours, the only time you get personalized instruction is during supervision where your example sheet is reviewed and you can ask questions. Math stackexchange if needed can help us here.

Here is the schedule for 2023/2024s. Every year the tripos publishes their curriculum and sometimes professors post lecture notes on their personal sites or by students like here or Neel Nanda's notes. Here is a sample recorded lecture.

• Mechanics
• Numbers and Sets
• Groups
• Vectors and Matrices
• Differential Equations

• Analysis

We will already know undergrad analysis and will have been exposed to measure from The Calculus Integral so instead we'll take Axler's free book Real Analysis.

• Vector Calculus

In addition to books/notes Cambridge uses we can take Sussman's book Functional Differential Geometry covering all the required theorems like Stoke's and Tensors and make them into programs using scheme.

• Probability

Axler's book already teaches us some probability w/measure and The Grimmett & Stirzaker book 4th ed recommended by Cambridge is like the bible of uncertainty modeling it covers everything plus has an accompanying work book One Thousand Exercises in Probability.

One of the most interesting courses you will ever take is imprecise probabilities and their 2019 book Game-Theoretic Foundations for Probability and Finance it covers betting, decision making, uncertainty. It uses game theory as a foundation for probability instead of pure measure theory.

• Dynamics and Relativity

On Prof David Tong's page

• Variational Principles (examined next year)

Very good lecture notes and this is what Sussman uses for his book SICM which we'll do once we get to Part 1B and take classical mechanics. This material is examined the next year in Part IB.

• Computational Projects
• Exams

All are here and they test both Michaelmas and Lent courses.

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