Notes on Intuitive Mathematics

[xavra42]

I came to these forums with the question how do I understand math intuitively. I joined and started this topic https://www.physicsforums.com/showthread.php?t=614527. I wanted to be able apply the methods quickly, without any calculation, so I could gain some further understanding. I read a few books, but they didn't contain the exact answer I was looking for. I ended up giving up and reading a seemingly unrelated book instead called "Thinking Fast and Slow". It's a cognitive science book that had a few profound implications

• We cannot apply mathematical knowledge without making a conscious effort
  I thought if I learned math it could run in the background, so to speak, and give me useful insights. I wasn't looking for exact answers, just broad behaviors that may be manipulated.
• We are susceptible to the same mistakes as the uneducated
  In the book, the author gathers professional statisticians and asks them basic questions that any high school statistics student could answer. The questions could be solved in your head by using basic stat concepts. Almost all of them got the answers wrong. The problem was not even the experts used their knowledge. Since that part of the brain isn't accessed when solving the problem it didn't matter that he knew it or not. This is troublesome because it greatly restricts application to problems where you are already mathematically focused (priming effect). For most people this only includes their job.
• True intuitive thinking like simple comparisons or reading someones expression is impossible
  As the book states, fast and intuitive thinking only includes a predetermined set of mental operations. None of these include any form of critical thinking whatsoever.

TL;DR For those who believe that you can get a feel for mathematics (like I did) that can be applied to everyday situations you are wrong. The brain is lazy (for evolutionary purposes) and only grants you access to that knowledge through conscious effort.

 

[DonAntonio]

I came to these forums with the question how do I understand math intuitively. I joined and started this topic https://www.physicsforums.com/showthread.php?t=614527. I wanted to be able apply the methods quickly, without any calculation, so I could gain some further understanding. I read a few books, but they didn't contain the exact answer I was looking for. I ended up giving up and reading a seemingly unrelated book instead called "Thinking Fast and Slow". It's a cognitive science book that had a few profound implications

• We cannot apply mathematical knowledge without making a conscious effort
  I thought if I learned math it could run in the background, so to speak, and give me useful insights. I wasn't looking for exact answers, just broad behaviors that may be manipulated.
• We are susceptible to the same mistakes as the uneducated
  In the book, the author gathers professional statisticians and asks them basic questions that any high school statistics student could answer. The questions could be solved in your head by using basic stat concepts. Almost all of them got the answers wrong. The problem was not even the experts used their knowledge. Since that part of the brain isn't accessed when solving the problem it didn't matter that he knew it or not. This is troublesome because it greatly restricts application to problems where you are already mathematically focused (priming effect). For most people this only includes their job.

• 
  $${}$$
  I think the main problem here is that statistics is not mathematics and thus statisticians are not mathematicians, in spite of what they may believe

  [*]True intuitive thinking like simple comparisons or reading someones expression is impossible
  As the book states, fast and intuitive thinking only includes a predetermined set of mental operations. None of these include any form of critical thinking whatsoever.

TL;DR For those who believe that you can get a feel for mathematics (like I did) that can be applied to everyday situations you are wrong. The brain is lazy (for evolutionary purposes) and only grants you access to that knowledge through conscious effort.

Ok...so? One of the most important things one learns after only 2-3 months in undergraduate school in mathematics

is NOT to rely at all on intuition...to do actual mathematics .

Intuition is fine to have a feeling about something, but we must always base that huntch on formal mathematics, otherwise

we could be delluding ourselves.

DonAntonio

 

[xavra42]

I think you've missed the point. Most mathematics enthusiasts will tell you that math can be applied to anything. I am just added on an important constraint set by your brain that states only when it is primed. As I said for most people that is only on the job and thus it is not very applicable in everyday situations. This is a huge drawback for me and one of the reasons I got into math to begin with.

 

[jgens]

As I said for most people that is only on the job and thus it is not very applicable in everyday situations.

What types of everyday situations are you talking about? Aside from high school level mathematics, there seems to be very little that is obviously applicable to everyday situations, so I am confused about what your criticism really is.

 

[xavra42]

What types of everyday situations are you talking about? Aside from high school level mathematics, there seems to be very little that is obviously applicable to everyday situations, so I am confused about what your criticism really is.

The higher level the math the less applicable in general. But statistics and calculus can give you insights in everyday events. Like the expected value from a gain with a risk attached to it, or understand how fast something increases using derivatives. I have used these often, esp when I am on the computer and i tab over to wolfram. The application IS there, the problem is that you also need to know when to use it which is a weakness of our minds.

 

[jgens]

So you're upset because you can't do things like compute expected values or derivatives more or less subconsciously?

 

[xavra42]

So you're upset because you can't do things like compute expected values or derivatives more or less subconsciously?

You don't need to compute to find out useful information. The problem is that it can't be used on the go, even if it is just concept based thinking.

 

[Obis]

I'm a math student, planning to continue my studies in the field of (computational) cognitive science, so I'm also interested in such matters. I'll share some thoughts.

First of all, the reality you see around you is not objective reality, it has already been structured by your brain, subconsciously. Lines, shapes, relations between objects, even objects themselves are more or less your brain's creation. It depends on the definition of mathematics, but, in some sense we do use mathematics when we apply structure to the unstructured information coming from the senses, subconsciously.

Secondly, there's a subtle issue with "The higher level the math the less applicable in general.". I'll try to illustrate the issue with an analogy.
If your goal was to analyze every chair in the world, you could simply analyze each of them, one by one. However, it would probably be more effective to find the properties that are shared by every chair, construct an "abstract chair" having those properties and analyze them. Such a chair doesn't exist in "reality" (it has no particular color and shape). You could continue and find the properties shared by any type of furniture - chairs, tables, beds, etc. Such an object is even more farther from "reality". However, is it correct to say that analysis of such an object is less applicable than the analysis of one particular chair?

 

[xavra42]

I'm a math student, planning to continue my studies in the field of (computational) cognitive science, so I'm also interested in such matters. I'll share some thoughts.

First of all, the reality you see around you is not objective reality, it has already been structured by your brain, subconsciously. Lines, shapes, relations between objects, even objects themselves are more or less your brain's creation. It depends on the definition of mathematics, but, in some sense we do use mathematics when we apply structure to the unstructured information coming from the senses, subconsciously.

Secondly, there's a subtle issue with "The higher level the math the less applicable in general.". I'll try to illustrate the issue with an analogy.
If your goal was to analyze every chair in the world, you could simply analyze each of them, one by one. However, it would probably be more effective to find the properties that are shared by every chair, construct an "abstract chair" having those properties and analyze them. Such a chair doesn't exist in "reality" (it has no particular color and shape). You could continue and find the properties shared by any type of furniture - chairs, tables, beds, etc. Such an object is even more farther from "reality". However, is it correct to say that analysis of such an object is less applicable than the analysis of one particular chair?

Not really what I was talking about, but since you are the only one responding why not :D Abstraction and in particular heuristics are the **** when applying mathematics. In the book that I talked about your brain is really good at averaging values like, for example, the length of various lines. But it has some blind spots so to speak where it is very terrible at summing values ( values can be length of even emotions). The long list of surprising limitations caused me to lose faith in the application of mathematics outside of my job and to create this thread about it.