Our Mind Can't be Measured
“The set of objects that we can think about and describe to others is limited from the start by our own humanity”. –Paul Lockhart.
The greatest challenge that we have as a human race is to overcome our physical and mental limits. We are limited in our thoughts and notions of a lot of things, conceiving infinites for example. However, what I found really amazing is that we can know what we can’t know. We can determine what things, objects or notions we can’t explain or understand. The most outstanding fact is that humans don’t give up; they create tools to understand and unravel “The Great Book”, as Galileo would say. Nevertheless, there are certain conceptions that we’ll never understand completely. What I admire the most is how human curiosity has discovered, created and achieved great things that have made our lives easier and more comfortable. I mean we can fly on an airplane to the other side of the world in a couple of hours! Or we can even go to space now. Aren’t those two things enough to be grateful with human kind? Everything started around 2,000 years ago, well even before, but what the Greeks documented and discovered changed human kind in an indescribable way. But, how did the Greeks get to that point? How did they make geometrical proofs? Did they discover it? Did they create it?
This question has been on my mind since I had a notion what mathematics is all about: Is mathematics really out there and we just created the concepts? Or is it inside us? Lockhart states that mathematical reality is not out there, but inside our minds. Everything happens in our minds and each of us has a different mathematical reality. “Mathematical reality is not “theirs”- it’s yours. You have an imaginary universe in your head whether you like it or not. You can choose to ignore it, or you can ask questions about it, but you cannot deny that it is very much a part of you”. (Page 398) This thought makes a lot of sense, however, I think that mathematical reality exists both inside our minds and outside them. We make a connection between our “inner world” to the “outer world”. We understand the outer world with our minds. And here we go back again into the problem I was stating at the beginning, our mind is limited and The Great Book isn’t. Well maybe it is, but till today we haven’t really discovered where the end is. To me, mathematical reality is yours, theirs and ours. I think that with this on mind is how great mathematicians have been building upon others thoughts and discoveries.
What is measurement? Is it something tangible or imaginary? “Measuring is relative”. – (Page 32) It is relative because it depends on what measuring unit we choose, it could be meters, centimeters, cups, etc. The word measurement goes beyond that because we are not dealing only with the exact measure but also with the concept. When we are measuring something we are comparing it to something else. “When we make shapes or patterns and measure them, we can choose any unit that we want to, keeping in mind that what we are really measuring is a proportion”. – (Page 33) After all, what we really care about is about proportions. That’s why we do it. Measuring is about making relations between objects, shapes or figures. It’s about understanding something imaginary, or it can be physical, and describing it with our language. It should be mentioned that our language is also a limit or a handicap to express how we understand reality because as Koestler would say, there are over precise words that don’t really nail what we want to express and language is always changing. “True creativity often starts when language ends”. – Arthur Koestler. This quote goes along very well to what Lockhart say “Measuring the diagonal of a rectangle requires insight and ingenuity; measuring the diagonal of a piece of paper is easy- just get out a ruler”. - (Page 49) So, in a way, language, units or something that is established sort of destroys our ingenuity to solve problems and find answers for ourselves. It’s more easy to use what’s given than to find an elegant and simple explanation.
“Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively. The truth is, I don’t know of any human activity as demanding of one’s imagination, intuition, and ingenuity”. – Paul Lockhart
The greatest challenge that we have as a human race is to overcome our physical and mental limits. We are limited in our thoughts and notions of a lot of things, conceiving infinites for example. However, what I found really amazing is that we can know what we can’t know. We can determine what things, objects or notions we can’t explain or understand. The most outstanding fact is that humans don’t give up; they create tools to understand and unravel “The Great Book”, as Galileo would say. Nevertheless, there are certain conceptions that we’ll never understand completely. What I admire the most is how human curiosity has discovered, created and achieved great things that have made our lives easier and more comfortable. I mean we can fly on an airplane to the other side of the world in a couple of hours! Or we can even go to space now. Aren’t those two things enough to be grateful with human kind? Everything started around 2,000 years ago, well even before, but what the Greeks documented and discovered changed human kind in an indescribable way. But, how did the Greeks get to that point? How did they make geometrical proofs? Did they discover it? Did they create it?
This question has been on my mind since I had a notion what mathematics is all about: Is mathematics really out there and we just created the concepts? Or is it inside us? Lockhart states that mathematical reality is not out there, but inside our minds. Everything happens in our minds and each of us has a different mathematical reality. “Mathematical reality is not “theirs”- it’s yours. You have an imaginary universe in your head whether you like it or not. You can choose to ignore it, or you can ask questions about it, but you cannot deny that it is very much a part of you”. (Page 398) This thought makes a lot of sense, however, I think that mathematical reality exists both inside our minds and outside them. We make a connection between our “inner world” to the “outer world”. We understand the outer world with our minds. And here we go back again into the problem I was stating at the beginning, our mind is limited and The Great Book isn’t. Well maybe it is, but till today we haven’t really discovered where the end is. To me, mathematical reality is yours, theirs and ours. I think that with this on mind is how great mathematicians have been building upon others thoughts and discoveries.
What is measurement? Is it something tangible or imaginary? “Measuring is relative”. – (Page 32) It is relative because it depends on what measuring unit we choose, it could be meters, centimeters, cups, etc. The word measurement goes beyond that because we are not dealing only with the exact measure but also with the concept. When we are measuring something we are comparing it to something else. “When we make shapes or patterns and measure them, we can choose any unit that we want to, keeping in mind that what we are really measuring is a proportion”. – (Page 33) After all, what we really care about is about proportions. That’s why we do it. Measuring is about making relations between objects, shapes or figures. It’s about understanding something imaginary, or it can be physical, and describing it with our language. It should be mentioned that our language is also a limit or a handicap to express how we understand reality because as Koestler would say, there are over precise words that don’t really nail what we want to express and language is always changing. “True creativity often starts when language ends”. – Arthur Koestler. This quote goes along very well to what Lockhart say “Measuring the diagonal of a rectangle requires insight and ingenuity; measuring the diagonal of a piece of paper is easy- just get out a ruler”. - (Page 49) So, in a way, language, units or something that is established sort of destroys our ingenuity to solve problems and find answers for ourselves. It’s more easy to use what’s given than to find an elegant and simple explanation.
“Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively. The truth is, I don’t know of any human activity as demanding of one’s imagination, intuition, and ingenuity”. – Paul Lockhart