Knowledge has always been advanced in human culture based on the ideas of others. Our entire knowledge structure today is based on what we, as a species, learned in the past. Each generation learns what the previous generation already knew, and then expands upon this base of knowledge for the next generation.

A problem with this system is that the amount of knowledge one must know before one can make an original contribution to the existing knowledge base increases with each generation. In other words, each generation spends more time than the previous generation learning about existing knowledge before adding their knowledge to the pool.

One way we have already begun to combat this problem is with increasing specialization. Instead of trying to learn everything from the previous generation, each individual learns only what is necessary in order to be able to advance the knowledge base and most individuals do very little to advance the knowledge base themselves, but instead provide the necessary support for our knowledge based society.

Here’s an example of the problem with specialization based on the field of mathematics. It used to be that almost every mathematician was an amateur without a lot of formal university training. **Euclid’s Elements** was a textbook for mathematicians for about 2000 years. Why was this possible? Well because quite simply, there wasn’t enough mathematics to be learned that you couldn’t contain a significant chunk of it in one book. So being an amateur mathematician was possible because you could read a few books about mathematics and suddenly be able to produce original ideas.

Now there are hardly any successful amateur mathematicians although many people still dabble in their spare time in mathematics. In this case, I define a "successful" mathematician as someone who has in some way advanced the pool of mathematical knowledge. The lack of amateur mathematicians is largely due to the fact that in order to be able to advance mathematics, one has to know quite a lot of mathematics, more than is really possible for the typical person. I can’t pick up a few books and suddenly be at the edge of what is known, instead I need years of training before I will reach that point, especially in the field of mathematics. Most of what we teach at the high school level, for example, is mathematics that was invented more than 300 years ago.

We have essentially hit the limit for what an amateur mathematician is capable of producing. We should expect only highly specialized mathematicians will produce new knowledge in the area of mathematics for the rest of our future as a species. This limit will eventually increase so that eventually no one will be able to add to the field of mathematics.

Increasing specialization can only take us so far in allowing us to keep increasing the knowledge base. Humans are an insatiably curious species, so it is far to assume that for most of us, increasing what we know as a species is a worthwhile goal. So what are we to do when it consumes an entire human lifespan just to learn enough to be able to add a small piece of knowledge to what we understand?

There are still a few areas where one can begin to add to existing knowledge without an enormous amount of investment in time learning the existing knowledge base. Interestingly enough, one of these is education itself. If we measure the complexity of a subject by the average number of years one needs to go to school before one can add to the existing knowledge base, then the field of education would be considered fairly simple. You can go to school for a mere 5 or 6 years after high school and be able to enter a classroom and learn about how people learn first hand. Add a year of learning about how to do research in the field of education and suddenly you can become someone who adds new knowledge to the pool of what is known about education.

Why is this true? Well, quite simply, as a species we are still mostly in the dark about how we learn, and what the best methods are for helping students learn. We have many theories about how learning works, and how to best apply it, but none of them has emerged as a definitive "best" theory.

Our ignorance as a profession of how people learn is astounding to me. It simply amazes me that we are still having a debate about whether having groups or not is best for learning. We still wonder if the introduction of technology in the classroom is worth it. Should kids be streamed or not? Is assigning homework right? How much homework is best? Home-schooled or not? Remember that we are the same species which is capable of sending someone off of our planet and then bringing them back, and that we did that more than 40 years ago! Why can’t we figure out how to make our education system work for everyone?

Another way to improve the odds of any individual person adding to the knowledge base besides increasing specialization is to greatly improve the efficiency of their education. Even a small improvement in the speed at which people learn the existing knowledge base could lead to years of extra time as a productive mathematician for example. If we knew more about how people learned, we should be able to translate this knowledge into improved opportunities for learning more about how the universe works, simply because we would be providing more time for specialists to work in their chosen field.

Furthermore, many people never have the opportunity to even consider adding to the pool of knowledge because they end their own education out of boredom! How many geniuses have we lost because of the way we constrain people so much in our system of education? Just improving graduation rates and allowing more personalization of education could do a lot to improve the efficiency of education.

We should be investing in our education systems more. We should invest heavily in research in education because that is an area where we can actually make an enormous improvement in the quality of education and eventually in how much we know as a species. A small gain in improved efficiency of our education system could lead to a large gain in end research because of the exponential effect of knowledge acquisition.

## Keith says:

It was the formalization of set theory, and previous problems in terms of set theory, beginning in the early twentieth century that made mathematics incomprehensible to many amateurs trying to learn it without formal teaching. There is a dire need to simplify group theory, algebraic geometry, and general abstract algebra so that a student with limited exposure to these areas, but with understanding of differential equations and linear algebra, could learn them.

August 10, 2011 — 11:43 pm

## Anonymister says:

I don’t think amateur mathematicians will miss much by not being on the cutting edge of research. In my experience, mathematics today is a pile of cr@p.

Pick up any current mathematics journal and you will see page after page of complete nonsense, theory after more esoteric theory that has absolutely no bearing on the real life. Mathematicians today just are concerned about defining weirder and weirder spaces, some meaningless mappings on them and then study their behaviour! The whole thing is not much dissimilar from games like War of Warcraft where a large number of people create their own fantasy world. The obvious difference is that those video games enthusiasts don’t live off your and my tax money.

Mathematics had a noble purpose once. Until the early 20th century, its purpose was to explain the workings of the physical world. Today the purpose of mathematics is to enable a bunch of people to live in a fairytale land – sheltered well away from the real world – and get their meaningless PhDs, write their meaningless fantasy papers, move on from lecturership to associateship., etc – all paid for by tax revenue.

December 17, 2011 — 2:23 am

## David Wees says:

It’s probably worth noting that some of the fanciful mathematics discovered/created a couple of hundred years ago had no apparent purpose and did not describe any portion of the real world. With the discovery of quantum mechanics, all of a sudden, this abstract field of mathematics had to be learned by physicists so that they could understand particle and quantum physics. It may be that some of the abstract knowledge mathematicians are advancing is necessary to describe as yet unknown physical phenomena.

April 23, 2012 — 7:50 pm

## Anonymous says:

I agree in large part with the last comment.

Based on experience, I know that there is a great chasm between advancing mathematics as an ‘amateur mathematician’ and getting it passed peer review which is made up of a body of people more intellectually similar to politicians than mathematicians. I suspect that there are many successful amateur mathematicians -successful simply meaning demonstrating new statements to be true using rigorous inferences- that are simply unknown because the academy has decided what is and ins’t acceptable work.

April 23, 2012 — 2:42 pm

## David Wees says:

Peer review should have more gradiations, from the formal review required to add the mathematical knowledge to the existing pool of tried and tested mathematical tradition, to comments on a blog post that provide feedback to the poster. This may alleviate some of the problems of politics, as you describe.

April 23, 2012 — 7:51 pm

## Stuart Anderson says:

I think mathematics is wide open for amateurs. It is true that to reach the frontiers of current knowledge in maths one must process a great body of knowledge, and one must specialise, but on the other hand, we have computers and software which can allow amateurs easy access to areas which involve complex calculations, and the internet provides many diverse sources of information for those who wish to go deeper.

Amateurs cannot reasonably expect to reach the same level as the best professional mathematicians. To be the best one has to devote a lifetimes work as well as being gifted with great ability. Its always been that way, an amateur could not hope to surpass Euler or Gauss or Poincare or Hilbert

For Recreational Mathematicians, things have never been better, there is a whole array of problems and areas one can work on without a lot of assumed knowledge. In certain areas such as tiling, amateurs are still making important discoveries.

July 7, 2012 — 9:14 pm

## Silas Davis says:

Whilst I agree with the sentiment that mathematics is somewhat unique when compared to sciences (or amongst sciences if you are that way inclined) such as physics in that mathematical knowledge tends to remain true once proven once and for all, rather than being subsumed and replaced by better theories, or indeed overthrown by them, as in physics, I think your assessment that eventually no-one will be able to contribute to mathematics is overly gloomy or at least very premature.

A significant factor reducing the learning-work necessary to reach the forefront of mathematics are novel generalisations, condensations, and refactorings of existing knowledge into more readily absorbable less cognitively dense pieces of work. It is far for clear how far such activity might go in making the boundaries of mathematical knowledge more accessible. I would cite modern language of algebraic geometry, dirac braket notation, or feynman diagrams for integration as examples of these types of innovation. Another factor worth considering is that if the trend of increasing life expactancy elogates the traditional stages of our life those that are compelled to study mathematics may find it more palatable to devote a longer period of time to it without that necessarily representing a greater fraction of the time they will have lived.

Regrettably specialise must have a role, but with that advancement we see some unification or at least connection. Consider points of contact between infinite graphs and topology for example; sometimes structures being studied have non-obvious morphisms. Maybe we’ll all just start studying category theory…

August 5, 2012 — 2:57 pm

## Ronald L. Altic says:

I am an amateur mathematician. I have been interested in numbers for many years; since before the 1980’s. Throughout those years I have made a number of discoveries. Many of them may have been discovered by others before my time. However, these discoveries are part of my memoirs during my work life as a machinist and as an amateur mathematician.

I don’t believe that my most important discovery is known. I have searched the internet and have perused many articles and texts and have not found anything that approaches my discovery.

I have two goals and/or purposes for writing a book.

One purpose is to encourage other amateurs to pursue their fascination with numbers no matter what their math background. I have had very little formal training in mathematics. The only course I had taken in high school was

. I have never taken a course in Algebra or Trigonometry or any of the higher math courses.Plane GeometryThe other purpose is to describe my most important discovery with the hope that one or more professional mathematicians will be curious enough to look at it.

The title of the book

may peak the interest of both amateurs and professionals.For Amateurs OnlyNovember 11, 2013 — 1:13 pm

## David Binner says:

Just came across this post and couldn’t resist replying.

I consider myself an amateur mathematician (and computer programmer.) I agree with your logic: the field of Mathematics has become so specialized you’d think a person needs a PhD in order to understand the cutting edge of

thatparticular field of research before they can even begin to add their own contribution to the field.On the other hand, I have often thought that the greatest advances are also the ones that are the easiest to understand. If a concept cannot be explained in terms that a high-school or university undergrad student can understand, it is not all that ground-breaking. In other words, I think most work done by PhDs are simply tweaks to the body of knowledge that only another PhD can understand; such work is confined to a very specialized sphere. If something really significant is discovered, something that shakes up the existing body of knowledge, it will be so far-reaching that even the lower fields of mathematics will be affected: i.e. – work at the cutting edge will again be reachable by amateurs.

A couple other comments:

1) The work of amateurs is rarely published, so their contributions usually go unknown. Unless a person is affiliated with a university, their work goes unnoticed in the field. Perhaps somebody, somewhere, made a significant discovery, but took the knowledge to the grave because they never had an opportunity to publish. Nowadays, we have blogs and websites on which to share our info but, even so, do amateur mathematicians make the effort to post their work online?

2) I think one way in which amateurs can play a role is to fill in some of the “gaps” in the existing body of information. For example, somebody working on an un-related problem, or approaching a problem from a completely off-the-wall direction, may stumble upon a solution to an as-yet unknown integral or ODE. The CRC Handbook has a pretty extensive list of integrals for which there are no exact solutions. Perhaps some of these could be found by amateurs. Yet there is no resource that coordinates workers. The Clay Math Contest offered prize money to people who could solve some outstanding math problems. Perhaps we need a site on which people can post a list of outstanding problems and others can post a list of solutions they have come across. Start a dialogue like the following:

Person A: “Hey everybody. In the course of my work, I found an exact solution to IntegralA.”

Person B. “That’s great! I have been seeking a solution to IntegralA. In fact, you have found a solution nobody knew about. You should send it to the appropriate math body, get it published, etc.”

At this point, that dialogue doesn’t even seem to be taking place. Where does an amateur go to see if some aspect of his work would be useful to the greater math community?

December 28, 2014 — 6:59 pm

## John says:

I do math in my spare time, and I fully agree. I am completely overwhelmed over the mass of prerequisite knowledge needed to seriously pursue any particular, rather specialized field of mathematics. David is also right in saying there are scarce resources for anyone to use if they wish to introduce their work to other mathematicians. And I like that you talked about math from hundreds of years ago being seemingly useless until it was needed to describe real physical phenomena discovered more recently. This is a really cool post; very thought provoking.

December 31, 2014 — 2:21 am

## ST says:

Why do we train for marathons? Learn to play musical instruments? Learn foreign languages? Do any of the commentators think they are going to win a marathon? Play in an international orchestra? Does this deter you from doing it? No. Study of math may have more positive effects on society then just utility in the way many of us think of it i.e. study of planetary motion, spread of diseases etc. How it preserves your mind and your health? helps you to build bonds with younger family members and so on may all be just as important.

March 26, 2015 — 4:30 pm

## commoner says:

I beg to differ. In saying that an amateur cannot contribute to mathematics you are limiting yourself and others. It is an arrogant and close minded view.

http://www.physforum.com/index.php?showtopic=117601

January 25, 2016 — 4:30 am

## David Wees says:

I don’t know how it’s arrogant to

wonderif amateurs have anything to contribute. It’s certainly arrogant to assume that they do not. I think you should reread my post as posing a question rather than asserting a truth.January 25, 2016 — 5:14 am

## commoner says:

The article is titled: “The Death of the Amateur Mathematician” In the article you explain that:

“We have essentially hit the limit for what an amateur mathematician is capable of producing. We should expect only highly specialized mathematicians will produce new knowledge in the area of mathematics for the rest of our future as a species. This limit will eventually increase so that eventually no one will be able to add to the field of mathematics.”

That sounds very much like an opinion. My previous comment was directed towards a point of view and not the author directly. Don’t get me wrong, I appreciate the fact that it was written. I just don’t happen to agree with it. When we say that there is nothing more to add to a subject we cut ourselves off from being able to progress forward. It’s a simple truth, not an attack. Amateurs are just as capable of coming up with new knowledge as anyone else. They should not be discouraged.

January 25, 2016 — 8:17 pm

## John Morrison says:

I cannot agree with the notion that we have hit any kind of limit as to what a so-called ‘amateur’ mathematician can produce. While it is true that the ‘high priests’ of mathematics have created an almost impenetrable language which needs to be mastered just to read their writings, there is much that can be discovered using elementary methods. The bonus is that the ‘high priests’ of the subject have long abandoned certain areas of mathematics and so the way is clear for amateurs to make classic discoveries.

Here is but one example from my website:

The angle sizes of an acute angled triangle can themselves form the sides of a triangle.

This is ultra-simple to prove but it has some amazing consequences, e.g. acute angled triangles where the angle sizes satisfy Pythagoras identity.

February 19, 2016 — 1:59 pm

## David Wees says:

So here’s a different flavor of this question then: will we eventually hit a limit where amateur mathematicians cannot add anything new to the field of mathematics?

This is different than whether amateur mathematicians can discover and prove mathematical ideas that are new to themselves which I think will obviously always be true.

February 19, 2016 — 2:04 pm

## Larry Cornell says:

The rate at which we are approaching that event horizon must also be slowing down. It’s probably a logistic growth phenomenon. The Internet is going to help for a few reasons. First silos of information are being destroyed. Second, the best teachers can share their best stuff with everyone in the web. Third, education will be essentially free to future generations. This will push things off for a while. A greater cooperation between AI and humans will also takes us a bit further. But eventually, we will hit the wall.

October 10, 2019 — 3:14 am