I watched the new Solo file on Sunday, that is the Han Solo film from Disney’s new range of films. I enjoyed it, the sci-fi-ness of it, the environments and scenery were cool. It's nice to be back in space and seeing fantastical concepts in motion. I also finally decided to invest in a new watch which had been a long time in the making. I finally settled on a Garmin forerunner 235 with integrated wrist-based heart rate monitoring. You can read about it here:Smash-run, Garmin 235 and various things. Its actually very nice and I’m very happy with my decision. So I’ve retired my long-serving and faithful Suunto Ambit 1 which was showing its age but was resilient.

I a couple of weeks back I added more functionality to my investment project, adding token-based authentication with JWT tokens as well as implementing the preliminary ideas behind role-based access. This now means that you can’t log in without signing up.

Also I added the first of my auditing functionality which keeps a record of the changes that have occurred throughout the investment predominantly but there are other audit activities such as ‘created a user’ etc. over and above the more useful activities within an investment such as ‘Changed value’, ‘associated factor x with investment’ etc that sort of thing.

That’s the login page, which protects all the pages unless a valid token is held by the user (having signed up and then logged in). The activity log that I mentioned looks something like this.

You’ll also notice that I’ve added a new top nav-bar which looks kinda nice although it doesn’t do much (except hold a logout button) but I think I’ll start moving more functionality into it moving forward.

There are some things that I’d still like to do but I’m as of yet uncertain how I’d like to do them. For instance, I’d like to add a value-system so that I can evaluate and compare investments. Particularly how values are performing. I’ve got some ideas about setting up a correlation matrix and/or Monte-Carlo simulation using investments stock history prices(time-scale end-of-day prices etc) but I’m unsure how or If I want to store this information in the SQL database. Also, There isn’t a consistent (and free) way to query financial information live across both UK and US stocks. So this is on the back-burner until I have a satisfactory plan of action.

I almost forgot I’ve also added per individual Group, Factor, Risk and Region relationships diagrams. What this means is that for a given one of those aforementioned entities, if you select one from any investment, it will show you all the other investments that also have this entity, pictorially – you could already see this relationship through the listing of related investments but I like graphs. 

Apart from that there are a few other inconsistencies that I’d like to address such as making navigation to and from pages that link to other investments more useful, for instance on the graph above, I’d like to that node links on the graph to be clickable and I’d like (possibly) to include a history of past navigations – but this isn’t really all that necessary and more a nice to have.

Also, I’ve started doing a series of blogs on Math fundamentals starting and my latest post being ‘What are fractions really?’. This is an effort to expose some of the underlying assumptions that most people don’t have when they talk about mathematics which is a problem I’ve encountered and I’m sure many have too – been hurried through math without due diligence. You can explore further what I mean in my mini-rant about the topic of general education of math in this article.  Most of my best insights so far has come from a book I’m reading called “Mathematics: Its Content, Methods and Meaning”.

A fraction by definition represents part of a whole.

The whole can be any number like 1 or a larger number can represent one whole such as 22 or 5 such that a fraction of 5/22 is a portion of the whole which is 22/22.

{jatex} 22 \div 22 {\jatex}

Usually 1 is represented as the whole but 22/22 or 7/7 is also a whole and so 25/22 and 2/7 are part of these wholes respectively.

In this way the denominator represents one whole part and the numerator a part of that whole part. Eg. As above there are 22 parts that make up a whole, when that whole is represented by 22.

A fraction represents a quantity and an amount just like a simple number does. However it represents a ratio of the part to the whole. That it is indicates a relationship between the part to the whole. A fraction can be represented as a decimal number. A decimal represents a notation and an amount that shows the portion of a whole.

A fraction represents an amount.

The amount usually represented by a fraction or a decimal is the amount left over when one number cannot equally go into another number. This occurs when things cannot be all dealt with in whole number outcomes. Eg. When one whole number cannot be divided equally by another whole number, the result is the amount of times it divided in(or goes in), plus the left over which is a portion of a whole unit that is left over that can’t go in fully any further as a whole unit but can as a part of a whole unit ie. a fraction of it.

When that number does not go equally into that other number a certain amount of times, the portion that does not fit is represented as a portion of the base interval or number

A fraction is a ratio, not directly a number in this form but it can be turned into a number or a decimal number. The fraction tells the story of how the numerator relates to the whole(which is the denominator). So 5 is an amount or quantity relative to the whole(13).

This ratio between numbers can be turned into a number which uses the two relations by defining how many times the numerator goes into the whole/denominator.
Thus the above fraction, the numerator is the focal point and seeing that it is not 13/13 which is a whole and rather 5/13 then it must go in 0 times because 13/13 would go in exactly once. So 5/13 represents a number smaller than 0 because it’s a portion of 1 so it must like between 1 and 0. It’s clearly not zero because that would be 0/13. Conveniently this can be worked out by dividing the part by the whole ie 5 divided by 13 which gives a number as 0.38461538 which is a portion of 1 and is between 0 and 1. So this number is the relationship that 5/13 has with 13/13 or 1.
It’s a part of 1 one but only the first 0.38461538 of it.

If 5/13 is 0.38461538 then 1/3 must be that number divided by 5 ie 1/3 is 0.07692308
Which means 13 of those must equal 1 and it does!

All fractions are about how an amount, represented by a fraction(which as the whole amount encoded in its denominator) relates to 1 - ie how smaller or bigger it is that 1.
This is key. Fractions should be considered as mechanisms to reason about 1(the whole) and parts of it.

 

 

I’ve been reading a very interesting and well written book lately which has re-kindled my appreciation of Math. Its called “Mathematics: Its Content, Methods and Meaning”. The great thing about it is that while it is a very large book, in fact its 3 bounded volumes – I’ve never come across an assessment of what actually mathematics is until now. It sounds dramatic but this really is crucial because I don't think many teachers really know how to rexplain what maths is. The main reason for this is because Math fairly abstract it is about representing abstractions, counting and measuring them and describing relations and rules amongst them. Doing things on abstractions is a fairly difficult concept to become comfortable with.

Typically speaking the concept of abstraction and thinking in abstract terms is something fairly higher-level and indeed the ability to do so has taken ages of development to master. No wonder its a difficult task to explain. Most of the time, the beautiful abstract nature of mathematics is lost(and not appreciated) and we’re told that mathematics is about adding up of numbers and counting and that two numbers when squared means this or that etc. This is so obtuse and disconnected for the true nature of mathematics and because of this it seems like a stupid exercise to have to be good at. Its like being annoyed.

The fundamentals of math that I’ve recently tried to encapsulate are that of its abstractness, that its really important to try grasp this about it as its actually more fundamental than say counting.

So I’ve worked out a good couple of train journeys now and I’ve worked out a good definition of what a number is and its not just a number as some high-school math teachers would have you believe.

The book goes thankfully describes what an abstract number is and why a number is a abstract idea. For all my efforts, I’ve determined that the following is the best way to think of a number as.

A number represents the occurrences of a concept.

This is quite amazing because not only does a number represent the amount or quantity of a concept, it can be any concept. That’s a plug on Math’s ability to be abstract in representing ideas. Speaking of ideas, that's really what a concept is, so numbers represent the amount/quantity of instances of the same idea.

A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev:

The abstractness of mathematics is easy to see. We operate with abstract numbers without worrying about how to relate them in each case to concrete objects.

  

Abstractions of this sort are characteristic for the whole of mathematics.

A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev | Location: 447

[Math] occur in a sequence of increasing degrees of abstraction,

Note:This is true, it seems the goal of maths is to find rules that govern abstractions so that the rules apply to everything that those abstractions can represent in the real world or by concrete things that are full of definition but share a common abstraction (like a base class)

Further more they talk about any maths being:

abstract concepts and their interrelations.

Now Arithmetic is the study of how numbers relate to each other, putting to a side the concept that the numbers actually represent. Numbers have properties, have patterns and through applying additional meaning to operations between numbers(quantities of concepts) such as addition, multiplication etc. we produce an output which is a story about numbers, but more importantly – its also represents the story of the ideas that those numbers represent/quantify. That's important. its not just about numbers and the dull, dry landscape of quantity having no relation to the concepts that they represent. I feel that we should always have in the back of our minds the concrete examples of abstract ideas that numbers represent. This is because you can then see the purpose and relationship that numbers have with your concept. Not merely that they are numbers.

Numbers are abstract representations of concepts, more precisely the amount of occurrence of those abstract concepts. They represent anything in Arithmetic. So its a study of numbers that can represent anything. Its quite abstract. So when you build a number on top of other numbers on top of operations involving number including patterns and laws – its easily possible to get lost in this wold of abstract thought without seeing the usefulness of what all these numbers and their results mean. In this way, you can cope with it if you can ‘see’ or bear in mind what these calculations are tell you about the ideas behind them.

They mean something in relation to the concepts that they represents. Many times we loose this, and become frustrated with memorising laws and get to a point where we find it hard to quantify something we’ve either forgotten(the conceptual ideas that numbers represent) or we were never told what they were to begin with and we started with numbers and that's it. That's dull. That’s the problem with modern interpretations of math.

A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev | Location: 606

arithmetic is relations among numbers.

  Arithmetic is linked to another mathematic discipline, namely Geometry by which is of measurement of figures and positions. The concept that you can fit something into another thing can be represented by numbers. That's what numbers are made for. This concept is called measurement. If you define a thing and say that it should fit into another thing, then you can quantify the occurrence of that concept. Thus you can represent it as numbers. In Geometry we take objects and abstract them to just an outline of them, we then measure them and we determine laws and patterns about this measurement of these geometry figures. In the same way a number abstracts the representation it is quantifying, geometry abstracts the thing its representing abstractly – without colour, smell or texture – just an outline.  

A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev says:

More generally, the concept of a geometric figure is the result of abstraction from all the properties of actual objects except their spatial form and dimensions.

This is so that whatever conclusions one makes about this outline, will apply to anything that this outline can represent in real life. But we do so by using numbers to do this. By counting the measurements of geometry lines and measuring positions and angles.

Position, angle and length are measurements that are represented by numbers. recall that numbers represent concepts. These are the concepts:

Length: The concept that something can fit into another thing.  We can count that.

Angle: The concept that one thing’s position is relative or has a relationship to another’s position. We can count that.

Position: The concept that one thing’s start or beginning point is relative or related to another things. We can measure that relationship.

This is how arithmetic(Numbers) represents measurement and how measurement is fundamental to the study of geometric figures.

Classic.

A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev | Location: 599

The subject matter of arithmetic is exactly this, the system of numbers with its mutual relations and rules.

Here are a few more quotes form the book that I determined to be useful about what Geometry is:

A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev | Location: 456

quantitative relations and spatial forms,

They go further and say:

abstracting them from all other properties of objects.

The Concept of the whole number is an interesting thing to figure out. Whole numbers have no parts, so they are not fractions which can compose whole numbers.

Fractions actually came about apparently according to the authors from measuring things and realising that things don’t go into things as an accumulation of equal parts. So they didn’t exist in arithmetic before Geometry which is interesting. The actual reason was that Pythagoras said that the diagonal of a square is made up from the relation of the two sides, but the relation ends up measuring to a number that is not something that can fit equally as a accumulation of whole lengths, such as 1. So the result would be somehwer between 1 and 2 for instance. This is obviously not 1 and this is obviously not 2 so you need another number to represent this quantity and this was when they invented fractions as a means to represent parts of the whole. These are also called irrational numbers.

Note:Basically maths is the rules that govern an abstraction. Relations make up those rules.

A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev | Location: 475

relation

Note:Relation here meaning known observation about the two abstractions

A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev | Location: 484

in some period of antiquity they represented the most advanced mathematical achievements of the age.

  

A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev | Location: 494

mathematics is significant only if the concrete phenomena have already been made the subject of a profound theory.

A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev | Location: 510

science proceeds to generalization,

To explain everything, you need to represent everything and this is what tending towards generalization is suggesting.

A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev | Location: 511

to formulation of laws and to mathematical expression of them.

  

A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev | Location: 559

inseparable property of a collection of objects,

   This  to talking about the property of quantity that collections share.

Operations such as addition and multiplication:

corresponds to placing together or uniting two or more collections,

of Maths:

discovered and assimilated the relations among the separate numbers,

Note:Indeed the discovery of relations are rules are what makes up the theorems that are base on those abstractions

Note:His is key, arithmetic is the study of numbers, how they relate to each other and thus their properties and general laws about that can describe these relations and properties. Arithmetic is Relationships and properties of an abstract concept of a number. A number is an abstraction as it represents any collection of a specified amount 

arithmetic is the science of actual quantitative relations considered abstractly,

  

mathematical designations in general: They provide an embodiment of abstract mathematical concepts.

This is important. In math, symbols represent a generic concept. It can be a very broad definition with many parts but be damn sure, it will represent something abstract and generic. Symbols are important because they can very quickly and concisely indicate what is meant to represent a mathematical concept ie a generalisation or abstraction