I have just spent all day listening to Norman Wildberger and universal hyperbolic geometry.
I feel swamped by the detail and the computational density of the material. But I think that I begin to see a glimmer of light.
The main issue for me is that the word algebra is used without real thought as to its meaning. But then the meaning of algebra is well illustrated by the way Norman presents it.
It is a kind of conundrum the fleetingly I can see the issue and the problem but then in a word Norman illustrates the solution. It is as if he intuitively knows what the meaning of the words that he is using. But then he uses the words as the standard practice insists. This is confusing.
I have a problem with the use of the word multiply, the word algebra, and the word geometry. In fact I had difficulty with the word arithmetic, due to the fact that it is derived from the Greek Arithmos. I do not sense that modern users of the word arithmetic or number have any conception of what Arithmos is. However they do have equivalents to the concept of the Arithmos which they do not recognise.
Of course, my own learning and understanding is constantly going under review. I have had to unlearn the information which I was taught and to learn this new information about the arithmos.
I am dictating this to my iPod. It is quite a new and unnerving experience to speak the thoughts that normally I would write down as I type. It is a learning process that I will have to get used to.
Many times the thoughts that I I have form as I write the word choice the emphasis all come to me in the action of writing. This is a different experience. As I speak the ideas come to me but the way to present it is a different way of doing business. Before I did not have to speak or pronounce my words correctly. Also the act of thinking was not disturbed by the sound of my voice. It is only with practice that
Having to think and speak is kind of mind breaking. I suppose when I am ready and I'm in the mood to speak, then this will become unnatural means of communication. I will perhaps become comfortable with this method of getting my thoughts down on paper.
But the more i practice and the quicker I speak the better I will be able to communicate. This device may be able to differentiate my speech and make this a much more enjoyable event.
When the words don't flow then there is this embarrassing silence, and then this pressure to say something but the embarrassment of not knowing what to say potentially that this could be very good and deliberating.
Norman has spent a great deal of time considering the very foundations of mathematics. .
I too have spent a great deal of time considering the foundations of mathematics. My conclusions are not yet complete, but I have been enlightened by many sources. Of these sources I have to say that Norman Wildberger has been the most influential. His thoughts and my thoughts have been very close in the past few months.
I have to recognise his consummate skill and dedication, his ability to communicate is outstanding. I also have to admit to learning and redacting conclusions that I have come to as well as changing opinions in the light of information that he presents.
Currently I am on a bit of a downer on algebra. It is because I have recognised that the Arabic word which means tortuous and convoluted thinking has been used inappropriately in the past. It has been used to create a new subject and in a sense to obscure arithmetic and to put geometry in the shade. However it does refer simply to the rhetorical style of those who are doing arithmetic or geometry.
This oratorical style is very demanding and is very mind numbing and torturous. That is why the Arabic word Al jibr was used to describe it. However it was never meant to be a different subject to geometry or Arithmetic. Historically however we can see that the use of symbols, the use of short notation, and the use of the actual equations to support discussion has taken over the meaning of the subject.
The words that were originally used to describe the notations were expressions. These expressions were quite simply used to describe the kind of thinking that the person was engaged they were very much mnemonic, that is to say that both the listener and the rhetor or speaker could concentrate fully in what was being discoursed or discussed. .
However later generations tended to concentrate on the symbols as if they were of magical significance. They failed to take into account the rhetoric that was required, the dynamic teaching that was required to communicate the information that the notation was simply reminding the speaker of. In fact they make a positive benefit of reducing the amount of rhetoric to the least possible, as if the symbols themselves gave greater clarity then the efforts of the speaker.
This of course is what happens when people become so wedded into a certain kind of jargon that they do not realise how much they are isolating themselves from normal speech. In so isolating themselves they also disconnect the general public who are around them. The result is that they fail to communicate that Worse still they begin to feel as if they are special. As if they have skill beyond the ability of everyday people. What they fail to realise is that they are not communicating effectively.
Not only are they not communicating with the people around them, they are also not communicating effectively with themselves. Therefore they become prone to making mistakes, which Even might seem like advances in knowledge. However when they come into some difficulty, that is when they give as if they have been flummoxed. The real difficulty is that they have undermined themselves and their understanding.
The history of arithmetic was meant to be one of man's intellectual achievement . It has always been presented as if it were a great achievement. Butlatterly it has come to be regarded as facile .
It is not facile, of course, it is in fact the basis of the symbolic thinking that commonly is called algebra. This is why the early 18th century and 19th century mathematicians worked so hard to try to extend the operations of multiplication addition division and subtraction into this symbolic type of arithmetic.
Where Hermann Grassmann made his greatest impact was in fact inmoving his father Justus Grassman's work about arithmetic and geomty fully into the symbolic arithmetic arena.