preventing students to think of math as lego bricks in which you can use a set of functions to build newer functions
Isn't this something that mathematical notation lets you be able to see at all? (I'm also not sure what you mean here exactly.)
By the way, have you heard what Euclid allegedly said to Ptolemy I when the king found the Elements to be difficult? "There is no royal road to geometry." Math is inherently challenging, and it's doubtful that removing powerful tools would make it any less so. I think of notation as being something that makes more complicated thoughts to be easier to have -- misquoting Alan Kay, good notation is worth 80 IQ points.
But surely notations vary a lot in quality. Think for example in roman numerals vs. positional number systems as alternative notations to work with arithmetic. One was surely a great advance over the other. So I wonder, what elements of our modern notations are akin to roman numbers, a suboptimal solution awaiting to be replaced with something vastly better. Is all perceived complexity in math inherent to it, or at least some of that is due to subpar notations?
Is all perceived complexity in math inherent to it
No, of course not, but I'd argue that every notation that's in use gives practitioners that use it strictly more mathematical capability. Notations make things that previously only geniuses could comprehend become things with wider accessibility.
Consider Roman numerals. Like anything, they are suboptimal, but without them, large numbers are essentially impossible to manipulate. They gave the business class the ability to record their finances and inventory, for example, which is a remarkable achievement. It's great when better things come along like positional number systems -- people can learn the art of division in grade school because of it, rather than needing to leave that to the experts -- but we shouldn't dismiss what was replaced as merely holding us back.
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Isn't this something that mathematical notation lets you be able to see at all? (I'm also not sure what you mean here exactly.)
By the way, have you heard what Euclid allegedly said to Ptolemy I when the king found the Elements to be difficult? "There is no royal road to geometry." Math is inherently challenging, and it's doubtful that removing powerful tools would make it any less so. I think of notation as being something that makes more complicated thoughts to be easier to have -- misquoting Alan Kay, good notation is worth 80 IQ points.
But surely notations vary a lot in quality. Think for example in roman numerals vs. positional number systems as alternative notations to work with arithmetic. One was surely a great advance over the other. So I wonder, what elements of our modern notations are akin to roman numbers, a suboptimal solution awaiting to be replaced with something vastly better. Is all perceived complexity in math inherent to it, or at least some of that is due to subpar notations?
No, of course not, but I'd argue that every notation that's in use gives practitioners that use it strictly more mathematical capability. Notations make things that previously only geniuses could comprehend become things with wider accessibility.
Consider Roman numerals. Like anything, they are suboptimal, but without them, large numbers are essentially impossible to manipulate. They gave the business class the ability to record their finances and inventory, for example, which is a remarkable achievement. It's great when better things come along like positional number systems -- people can learn the art of division in grade school because of it, rather than needing to leave that to the experts -- but we shouldn't dismiss what was replaced as merely holding us back.