Everything they taught you was wrong
"Everything they taught you was wrong"...
A biochemist said that to me recently: entering upon a PhD, remembering what had been said when she embarked upon A level, degree, MSc, PhD....
At each stage the new teacher, lecturer, tutor, supervisor .... everything they taught you was wrong, now we start to learn what is right.
We are all familiar with it - at various stages in our education, someone takes apparent delight in disillusioning us - everything we learnt is wrong, so we have to start again.
But the thing is, it wasn't wrong.
When we learn something - anything that is at all complex - we have to simplify it. Almost everything worth learning is unutterably complex - complicated almost or actually beyond understanding. So what we do is, to learn an aspect of it. 'Aspect' is a visual term - it describes the way you look at something: the view you take of it, how it looks from a given viewpoint.
Take a house for example, Not much more familiar, nor more simple, than a house. One of the first things we learn to draw: a box, a wedge shaped roof, four windows, a door, maybe a chimney. But that is not a house - it is a view of a house. The front view is called the 'front elevation'. We could also draw the side view - the 'side elevation' - although that probably has few or no windows so is a bit boring. Or we could draw the plan view - seen from above, maybe a rather dull rectangle with perhaps a smaller square for the chimney.
Each of these is a view - an 'elevation', an aspect. In the case of the drawings we are rendering a 3D object onto 2D paper. That is called 'reducing the dimensionality' of the object - from 3D to 2D. It is also, more formally, called a 'projection': which is quite illustrative actually - a projection is quite literally that - projecting like a shadowgram, the 3D house projected onto the 2D paper. You simplify the house by projecting it, so you can view and understand aspects of it. If you advance, you learn to draw an 'isometric' projection - a 3D view on 2D paper - a view of the house from an angle, and a bit above, that actually looks - even on flat paper - like a 3D house. That is quite clever - projecting three dimensions onto two but still retaining the three dimensionality. But we can't do that much - we usually have to reduce the dimensionality - to make a projection - in order to make sense of what we see.
Each of the views of a house shows you different things about it. The front elevation helps you to see where the doors and windows are: the plan view shows you its overall 'footprint': the isometric view gives you an idea of what it would look like in 3D reality.
If you make your projections transparent to a degree you can see more. Estate agents often provide 'floor plans' - as if the roof, or the whole top floor, has been ripped off so you can see where the walls are, and how rooms connect through doorways, and where furniture might go or people might walk to get from the kitchen to the back door to the garden. These are also projections, but with choices as to what is important - what is visible - and what is unimportant - what is not shown.
My biochemist friend was taught projections. At secondary school she would have been shown how molecules are like balls connected with sticks - how they can be drawn, flat, on paper, to show the connections between atoms and how those atoms are ordered in space. Those were projections - specifically, the flat 2D one is the 'Fischer' projection - designed to show how atoms are connected into molecules. The 'Haworth' projection might have come later - it is a 3D drawing on 2D paper - an isometric drawing, showing something of the 3D shape of the molecule. Later, projections might have shown the quantum mechanical electron orbitals that make the inter-atom bonds - like planetary orbits, or perhaps like foggy 3D blobs.
But a projection is not just a random drawing, it is a fundamental mathematical tool to visualize - to 'see' something about an aspect of something. Projections simplify, they draw out aspects in such a way that we can understand some things even about very complex systems. In Signal Processing, which is my own specialist field, a projection is a precise mathematical operation. If you want to be prissy about it, it is a 'vector dot product' or a 'filter' - but we won't get prissy, we will just accept that it is like a child's drawing of a house from the front, or above, or the side, or with the roof ripped off.
Projections - and their companions, rotations (which let you rotate things before you project, so you can choose the most appealing aspect) - are the life blood of signal processing. Actually they are pretty much all there is, tool-wise - we rotate, and we project, and by doing so we hope to gain insight. (The maths of the rotating and projecting is the easy bit, even though most people think it is hard - the insight is the hard bit).
You can project in weird ways too. For instance, suppose you took the double helix strand of DNA and sort of untwisted it, so it laid flat: that would be a projection - a twisted one, but perfectly valid, and helpful in its way.
And you can project more dimensions than three, too. Suppose we took a GPS (satellite navigation) data set - each data point is 4 dimensional - 4D - latitude, longitude, altitude and time. You can't readily visualize all four dimensions at once, so you project. Latitude and longitude flat on a map gives you the track, the path you followed: altitude against distance travelled gives you the hill profile up which you cycled; speed against time shows you how fast you ran. Each of these is a projection, sometimes with a rotation or a twist, but still a projection, a simplification, a reduction in dimensions.
In mathematical worlds we can model lots of things as 'dimensions'. I don't want to go there - fascinating as it is - but when a particle physicist stuns you into a vacant-eyed stare by telling you the world is '10-dimensional', all she means is that we can make a mathematical model where we could plot ten things at once: and we can only understand that by projecting two or three of them at a time. OK, maybe the 10-dimensional particle physics world is not as helpful as I hoped, but we can talk about that another time: just trust me, the world can have as many dimensions as you want but you can only see two or three at a time, so you have to project, to reduce the dimensionality to understandable proportions - to see things in perspective.
And that is what the teacher, and the lecturer, and the supervisor, are all doing: rotating and projecting, trying to gain - and offer - insights into aspects of the things they are teaching: bits of it, each at a time, so that you can step by step learn to understand more of its complexity.
We all know that a floor plan is not a house: we know full well that if we are choosing a house to buy we ought to look at it from front and side and back and from the street and from the garden and from inside and on a map and....
And that is what the lecturer meant when he said that everything the teacher taught you was wrong. That the teacher did not teach you everything. But that doesn't mean that what they taught you was wrong - any more than the house floor plan is 'wrong'. It means that what they taught you was an aspect - one amongst many - and with luck what they will teach you next is another aspect, insightful, illuminating, fascinating, into the incredibly complex multi-dimensional complexity of the real world.
So everything they taught you was right - but each thing you learnt is only an aspect of the whole.
Which is kind of cool, I think. And a lot better than it being wrong.
A biochemist said that to me recently: entering upon a PhD, remembering what had been said when she embarked upon A level, degree, MSc, PhD....
At each stage the new teacher, lecturer, tutor, supervisor .... everything they taught you was wrong, now we start to learn what is right.
We are all familiar with it - at various stages in our education, someone takes apparent delight in disillusioning us - everything we learnt is wrong, so we have to start again.
But the thing is, it wasn't wrong.
When we learn something - anything that is at all complex - we have to simplify it. Almost everything worth learning is unutterably complex - complicated almost or actually beyond understanding. So what we do is, to learn an aspect of it. 'Aspect' is a visual term - it describes the way you look at something: the view you take of it, how it looks from a given viewpoint.
Take a house for example, Not much more familiar, nor more simple, than a house. One of the first things we learn to draw: a box, a wedge shaped roof, four windows, a door, maybe a chimney. But that is not a house - it is a view of a house. The front view is called the 'front elevation'. We could also draw the side view - the 'side elevation' - although that probably has few or no windows so is a bit boring. Or we could draw the plan view - seen from above, maybe a rather dull rectangle with perhaps a smaller square for the chimney.
Each of these is a view - an 'elevation', an aspect. In the case of the drawings we are rendering a 3D object onto 2D paper. That is called 'reducing the dimensionality' of the object - from 3D to 2D. It is also, more formally, called a 'projection': which is quite illustrative actually - a projection is quite literally that - projecting like a shadowgram, the 3D house projected onto the 2D paper. You simplify the house by projecting it, so you can view and understand aspects of it. If you advance, you learn to draw an 'isometric' projection - a 3D view on 2D paper - a view of the house from an angle, and a bit above, that actually looks - even on flat paper - like a 3D house. That is quite clever - projecting three dimensions onto two but still retaining the three dimensionality. But we can't do that much - we usually have to reduce the dimensionality - to make a projection - in order to make sense of what we see.
Each of the views of a house shows you different things about it. The front elevation helps you to see where the doors and windows are: the plan view shows you its overall 'footprint': the isometric view gives you an idea of what it would look like in 3D reality.
If you make your projections transparent to a degree you can see more. Estate agents often provide 'floor plans' - as if the roof, or the whole top floor, has been ripped off so you can see where the walls are, and how rooms connect through doorways, and where furniture might go or people might walk to get from the kitchen to the back door to the garden. These are also projections, but with choices as to what is important - what is visible - and what is unimportant - what is not shown.
My biochemist friend was taught projections. At secondary school she would have been shown how molecules are like balls connected with sticks - how they can be drawn, flat, on paper, to show the connections between atoms and how those atoms are ordered in space. Those were projections - specifically, the flat 2D one is the 'Fischer' projection - designed to show how atoms are connected into molecules. The 'Haworth' projection might have come later - it is a 3D drawing on 2D paper - an isometric drawing, showing something of the 3D shape of the molecule. Later, projections might have shown the quantum mechanical electron orbitals that make the inter-atom bonds - like planetary orbits, or perhaps like foggy 3D blobs.
But a projection is not just a random drawing, it is a fundamental mathematical tool to visualize - to 'see' something about an aspect of something. Projections simplify, they draw out aspects in such a way that we can understand some things even about very complex systems. In Signal Processing, which is my own specialist field, a projection is a precise mathematical operation. If you want to be prissy about it, it is a 'vector dot product' or a 'filter' - but we won't get prissy, we will just accept that it is like a child's drawing of a house from the front, or above, or the side, or with the roof ripped off.
Projections - and their companions, rotations (which let you rotate things before you project, so you can choose the most appealing aspect) - are the life blood of signal processing. Actually they are pretty much all there is, tool-wise - we rotate, and we project, and by doing so we hope to gain insight. (The maths of the rotating and projecting is the easy bit, even though most people think it is hard - the insight is the hard bit).
You can project in weird ways too. For instance, suppose you took the double helix strand of DNA and sort of untwisted it, so it laid flat: that would be a projection - a twisted one, but perfectly valid, and helpful in its way.
And you can project more dimensions than three, too. Suppose we took a GPS (satellite navigation) data set - each data point is 4 dimensional - 4D - latitude, longitude, altitude and time. You can't readily visualize all four dimensions at once, so you project. Latitude and longitude flat on a map gives you the track, the path you followed: altitude against distance travelled gives you the hill profile up which you cycled; speed against time shows you how fast you ran. Each of these is a projection, sometimes with a rotation or a twist, but still a projection, a simplification, a reduction in dimensions.
In mathematical worlds we can model lots of things as 'dimensions'. I don't want to go there - fascinating as it is - but when a particle physicist stuns you into a vacant-eyed stare by telling you the world is '10-dimensional', all she means is that we can make a mathematical model where we could plot ten things at once: and we can only understand that by projecting two or three of them at a time. OK, maybe the 10-dimensional particle physics world is not as helpful as I hoped, but we can talk about that another time: just trust me, the world can have as many dimensions as you want but you can only see two or three at a time, so you have to project, to reduce the dimensionality to understandable proportions - to see things in perspective.
And that is what the teacher, and the lecturer, and the supervisor, are all doing: rotating and projecting, trying to gain - and offer - insights into aspects of the things they are teaching: bits of it, each at a time, so that you can step by step learn to understand more of its complexity.
We all know that a floor plan is not a house: we know full well that if we are choosing a house to buy we ought to look at it from front and side and back and from the street and from the garden and from inside and on a map and....
And that is what the lecturer meant when he said that everything the teacher taught you was wrong. That the teacher did not teach you everything. But that doesn't mean that what they taught you was wrong - any more than the house floor plan is 'wrong'. It means that what they taught you was an aspect - one amongst many - and with luck what they will teach you next is another aspect, insightful, illuminating, fascinating, into the incredibly complex multi-dimensional complexity of the real world.
So everything they taught you was right - but each thing you learnt is only an aspect of the whole.
Which is kind of cool, I think. And a lot better than it being wrong.
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