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