I’ve been a bit lazy with this website but I have finally completed my masters thesis which can be found below. I will try to publish a post which captures the essence of what I have done for my thesis and present it in a way that is accessible to non-mathematicians. Stay tuned.
Masters Thesis: Characterisations for the category of Hilbert Spaces
Category theory is an algebraic framework based on the composition of functions. Categories consist of objects and morphisms between objects. A dagger category is a type of category which has a notion of reversibility for each morphism. A monoidal category is one which allows the joining of objects and of morphisms in parallel, rather than in series as with composition. This joining is done in such a way as to satisfy certain coherence conditions.
The categories of real and of complex Hilbert spaces with bounded linear maps are dagger monoidal categories and have received purely categorical characterisations by Chris Heunen and Andre Kornell. This characterisation is achieved through Sol`er’s theorem, a result which shows that certain orthomodularity conditions on a Hermitian space over an involutive division ring result in a Hilbert space with the division ring being either the reals, complexes or quarternions.
The Heunen-Kornell characterisation makes use of a monoidal structure, which in turn excludes the category of quarternionic Hilbert spaces. We provide an alternative characterisation without the assumption of monoidal structure on the category. This new approach not only characterises the categories of real and of complex Hilbert spaces, but also the category of quaternionic Hilbert spaces.
Abstract Nonsense
“Why is a raven like a writing desk?”
The mad hatter hadn’t the slightest idea and Lewis intended it to remain unanswered, but the question does suggest that ravens and writing desks are similar. How would they be similar or even the same? In mathematics, this question is taken very seriously, not necessarily for fictional riddles by fictional characters but for fictional riddles by real people.
An example a question like this is,
“When are two sets the same?”
To be clear, a set just is a collection of things nothing more. The things could be anything, they could be all things in the observable universe, the counting numbers, the teams in the grand final but note that to describe what was in the set, we had to refer to the context of the things or what the things do.
But when are two sets the same? We could say they are the same when they contain the same things but then we would need to understand what sameness means for the things. As an example, consider the two sets 
The way mathematicians like to show two sets are the same is to show that each thing in one of the sets has a partner in the other. This would be called a one-to-one correspondence and is a kind of way sets relate to one another.
There are many ways two sets of different sizes can be related but one-to-one correspondences allow us to have a notion of sameness in the category of sets.
But what about the category of all universes or the category of grand finals? How do we talk about sameness here?
Introducing Category Theory. This the theory of mathematical categories and is way of looking at sameness in a different way. But first,
What is a category?
Categories are more about movement and how things relate whereas sets are static things in themselves. The definition of a category is as follows:
A category is a collection of objects,
and arrows between objects,
We can imagine the objects 
is the composition of with 
We can also make round trips there and back again, as if we didn’t leave at all.
is called the identity arrowThese identity arrows are important to our categorical idea of sameness explained later but the meaning of such non-movements is that moving from one object to another and then take staying in place is equal to just the first movement. The final rule is that we can take any path like in the diagram below
i.e. brackets don’t matter!One of the main points of category theory is that we can talk about properties of objects without referring to what the objects are directly. Now if I want to say two objects
and 
The idea is that these movements between objects carry along with them only the properties we care about. The technical term for such a movement
So why is a raven like a writing desk? We have a slight idea, though the answer will depend on which category containing collections of ravens and writing desks you’re referring to.
References:
Baez, John C., and James Dolan. “Categorification.” In Higher Category Theory, eds. Ezra Getzler and Mikhail Kapranov, Contemp. Math. 230, American Mathematical Society, Providence, Rhode Island, 1998, pp. 1-36.
Carroll, Lewis.Alice’s adventures in Wonderland .[New York, Boston, T. Y.Crowell & co, 1893]
Illustration by Shay Tobin
The Uniqueness of Natural Numbers
The following is the introduction and conclusion to the report I wrote during my course in pure mathematics at Macquaire University 2020. I’ve omitted the bulk of the report due to mathematical formatting complications while attempting to translate LaTex into WordPress. The full report can be downloaded at the end of this post.
One fish, two fish, red fish, blue fish. There is something odd about this sequence as we read it in our heads. Beginning with one fish and then following with two fish feels natural, since if you see one fish and then you see another fish, you’ve seen two fish. However the ordering of red fish and then blue fish seems arbitrary, Dr Seuss may as well have said blue fish, red fish and no one would have batted an eye. What would be strange is to say you’ve seen two fish and then note you’ve seen one fish. This is given in the statement that you’ve seen two fish, since 2 = 1 + 1.
Natural numbers have properties that may feel obvious or trivial but we will see that this collection of properties can be uniquely characterised, up to isomorphism, using a few primitive notions. At first I will discuss the characterisation of the natural numbers using the Peano axioms and introduce the recursion theorem. Then a recursive definition of the arithmetic of the natural numbers will be given. Then I will use the recursion theorem and inductive methods to prove that any model (N, s′, 0′) of the Peano axioms will be structurally isomorphic to the natural numbers.
In Turin, Italy 1889, the mathematician Guiseppe Peano developed a way to describe the arithmetic of numbers using the primitive notions 0, number and ’successor of’. He used the following five postulates;
(P1) 0 is a number.
(P2) If a is a number then the successor of a is a number.
(P3) 0 is not the successor of any number.
(P4) If two numbers have the same successor, the two numbers are identical.
(P5) If N is a class containing 0 and the successor of every number belonging to N, then every number belongs to N.
It should be noted that Peano himself did not include 0 in the set of natural numbers and instead used 1 as the initial number. Hopefully by the end of this report it should be clear that these two conventions are equivalent to one another in that they both satisfy the Peano axioms. In fact we could start at any integer. This is alluded to in a quote by Peano in Bertrand Russell’s Principles of Mathematics.
All the systems which satisfy the primitive propositions have a one-one cor- respondence with the numbers. Number is what is obtained from all these systems by abstraction; in otherwords, number is the system which has all the properties enunciated in the primitive propositions, and only those.
Our task will be to find this so called one-one correspondence in the form of a bijection which preserves the arithmetic of these numbers and to do this we will introduce the recursion theorem… . . .
. . . … some mathematical induction … . . .
. . . … We’ve now seen that φ : N → N’ is a bijection which preserves addition, multiplication and order between the natural numbers (N,0,s) and any model of the Peano axioms (N’, 0′, s′). We used (N’, 0′, s′) for could have used for (N, 1, s) like Peano’s original axioms or even (N, 42, s). The recursion theorem always makes sure there exists a unique isomorphism between any two models of the Peano axioms.
Note: I have uploaded document which includes the report in full, containing a proof to the recursion theorem and a construction this one-one correspondence between the natural numbers and any model of the peano axioms.
A simple proof that two wrongs make a right
Actually this is a simple proof that two negative numbers when multiplied make a positive number. ( Multiplication between 
We will prove that
Firstly we need to agree on a few rules
That final rule is called distributivity and is the key to understanding the following.
And we are done.
Maths for Many (part 1)
The purpose of this text is to attempt to express some of the most abstract and inspiring topics I have encountered during my studies in an informal and hopefully accessible way. The formality that surrounds mathematics can act as a barrier and even a repellant to many. Although there is a wealth of beauty behind them, they have not been raised by mathematicians for the purposes of hiding it. To start things off, we’ll jump straight into the topic of numbers.
What are Numbers?
This is a big question but one that can be particularly strange to answer. The first numbers we are introduced to are those we use for counting: “one fish, two fish, three fish,….n-fish?”. These are named the natural numbers. We can be clearer with what we mean by “one” or “two” however.
Firstly, the amount of planets you have lived on is also how many parts this series currently has at the writing of this blog.
| Amount of planets | = | Amount of parts |
The amount of shoes you buy is hopefully the same amount of feet you have which should also correspond to the number of legs attached. To be sure of this you can simply make sure every shoe fits a unique foot attached to a leg. We could give this property a name, “two”.
What we have done is created a class of sets that shares the property of having two elements. Similarly we could do the same for things that share the amount of planets you have lived on and name this class “one”. This notion of number is the class itself.
This description is quite nice intuitively and corresponds to how we are first introduced to numbers however what about all the things we can do to numbers, arithmetic for example or even some of the stranger numbers like
58209749445923078164062862089986280348253421170679
82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196
44288109756659334461284756482337867831652712019091
45648566923460348610454326648213393607260249141273
72458700660631558817488152092096282925409171536436
78925903600113305305488204665213841469519415116094
33057270365759591953092186117381932611793105118548
07446237996274956735188575272489122793818301194912
98336733624406566430860213949463952247371907021798
60943702770539217176293176752384674818467669405132
00056812714526356082778577134275778960917363717872
14684409012249534301465495853710507922796892589235
42019956112129021960864034418159813629774771309960
51870721134999999837297804995105973173281609631859
50244594553469083026425223082533446850352619311881
71010003137838752886587533208381420617177669147303
59825349042875546873115956286388235378759375195778
18577805321712268066130019278766111959092164201989…
What do I want to do?
What do I want to do?
I have been enjoying my education for the past two and a half years. It has perhaps been the most challenging task I’ve attempted and recently I have been in a state of flux, I am questioning my motives for why I study at all.
I love physics, so it pained me to leave it. I remember it as a core motivation for all the things I was doing and now I’ve excluded it from my studies, switching all my attention to “pure” maths. However, I am still motivated by physics as it connects these abstract worlds in which I find myself to something more real. I have also found its importance and beauty which is recognised by those I’ve met who work and live through it.
There is a feeling of absolute truth to the study of mathematics which physics simply cannot satisfy in me. This is in part for the reason of my current priority towards it or at least, my goal to give physics some level of certainty through mathematics. My curiosity about the world, of perception and of thought gives rise to this interest in mathematical physics, even towards the surface of mathematical logic. These are not unique curiosities and luckily for me, people through the ages have asked these questions and have collectively been working towards understanding their world.
As a kid, my curiosity was peeked looking at mechanics and astronomy and being amazed through my small personal discoveries. Electricity and magnetism were a very early wonder as I would constantly be making electric magnets from AA or 9 volt batteries, found wire and nails, not understanding the deeper underlying physics but understanding enough to build and test them. It wasn’t until high school that I was beginning to gain a better knowledge of how these amazing devices worked. I suppose I am still fascinated by them but as I learnt more and more my interests were directed at devices and ideas with greater perplexing wonder.
University (well a book I read along side a physics course) really showed me a deeper understanding to electromagnetism through Maxwell’s equations for which I only still have a fairly shallow understanding.
The natural world is both beautiful and mysterious and these are the stems of pleasures for the activities I subject myself to. Mathematics however has really opened a new world of possibilities, giving me a new lens to see reality and even now directing my attention to thought, reason and the self. This has proven to be the most difficult activity I have found myself in so far. It is testing my patience, attention and devotion to what I considered to be my core motivation.
I would like to be a scientist, a mathematician, a philosopher, an artist but these are titles and I have begun to think that to ‘be’ something is to pamper to an idealised notion of some category which fits all the positive associations of some specific people outside myself to a classifying name. For me, to desire to be these names is to lead myself away from what I want to give my attention and ultimately I see myself walking down the path of learning and expression.
Life, Art and Maths
The following is a blog written for my AMSI summer research 2018-19.
Throughout my life my mother has pointed out the curious and unique aesthetics of Australian trees, this may seem strange at first but my mother is an artist and teacher which may seem stranger still. It was this style of attention which developed my curiosity, a love for the oddity of nature and how seemingly simple things in the world can have complex mysteries to them. I remember first trying to draw a landscape, green trees with brown trunks, blue sky and a round sun sitting slightly to the side, a staple image on any parents fridge. As a mother would do, she gave me praise for the nice picture but then pointed out that this isn’t what trees looked like and referred me to a nearby gum. It had peeling orange and grey bark, some of which was reflecting the blue of the sky, winding branches which were impossible to track and most notably, it looked totally unlike the tree I drew. It was an important lesson, not only to improve my drawing but to also be critical of where my thoughts and interpretations of the world, are in conflict with its actual representations (these were not specifically the words I formed at the time).
Mathematics and science through schooling were subjects I enjoyed but like every subject, my passion and interest was teacher dependent. I was lucky to have devoted science teachers who would bare my questions after class, this pandered to my growing love for physics. The few less interested maths teachers on the other hand would tear down any curious thoughts, even set punishments for being distracted by the curious M.C. Escher posters on the wall however this would lead me to having more drawings in my maths books than maths. Unfortunately these teachers were abundant in my senior years of high school and I quit maths.
This forced me down the art path and I found myself studying design at university still curious about the world. My interest was peaked initially as it incorporated my love for art while learning the tools for technical application but when the professional world of marketing and advertising became the focus I started writing maths in my drawing book.
A few years and artistic endeavors later I had almost forgotten my passion for physics until the detection of gravitational waves was announced by LIGO. The notion of humanity gaining a new way to observe physical reality fascinated me so greatly, however my inability to tap into having greater understanding of these concepts became exceptionally apparent. I was lucky to have a particularly mathematically literate friend who lent me his general relativity textbook. It was wondrous, until I hit the 8th page. It was then I began to make the decision to study physics, but mathematics was necessary to do this.
I started from scratch in a humble maths bridging course, beginning with 2+2=4 up to introductory calculus and it was love at first sight, there was something so pleasing that simple ideas with careful reasoning lead to grand consequences. Mathematics continues to not only satisfy the knowledge seeking/puzzle solving side of my mind but has begun to indulge the creative/artistic parts as well and denotes the start of what is currently a tiny subset in the study of life, the universe and everything.
Writing in public
After little deliberation I have decided to start this blog. Like most things, its purpose is unclear. It should be known however that I intend to write anything that interests me or anything else. I cannot see many people seeing my writing and I will use this ignorance to write freely.
Shay's Maths and Art blog 