While I think that tools such as Matlab are useful, they are more harmful to students and researchers who use it blindly as Ahmad so rightly puts it. These tools are empowering to people that not merely know how to use it but more importantly  recognize what are their limitations.

Excellent question!  I personally think it can go down in we define all mathematics in MATLAB code, and don't carefully look at what's going on (at least the maths).  It really depends.  In my opinion at least, it depends on whether they understand what they are doing, or are just blindly using code that they don't quite follow.  It's the same analogy as with programmers and computer scientists.  A programmer may not understand why things work on a machine (e.g., why an algorithm is correct) but may know how to refine the code or use it without much thought.  We should look at things like MATLAB like tools: they are beneficial when we understand them, but become a crutch if you do not if you want to be relevant in research relating to that computing.

It's the same way with teaching mathematics: if you give everybody a calculator without understanding what they are doing, they won't "get it" but may know how to use the tool more often than the actual maths (which doesn't require the calculator).  Pen and paper is the way to go to learn it, and applying it (whether it be a calculator or MATLAB) is applying it.  I think in many regards as younger individuals get ease of access to technology, I do believe this dependency on using tools but not the understanding behind them (leaving them blind to how it actually works) will only grow as generally people are stubborn when it comes to learning things outside (in their view) their interests.  At the end of the day, no matter if researcher or not, they just want results and these tools will help them.  I think on what we can figure out is getting better, but access and understanding to get there is only getting harder if you aren't wanting to delve into the science of computing (e.g., Theoretical Computer Science).

our children are using calculators in schools and we are using MATLAB in our offices, what is the shape of the generations to come?

In fact, MatLab or other softwares are making users more able to do many statistical and mathmatecal functions.

I think it's great that we are finally offloading certain cognitive processes onto machines.  This tendency likely signifies another step toward better augmented cognition, meaning that we can leave complex calculation to software while improving our creativity, information search patterns, data management processes, and overall cognitive performance.

Inventing the wheel didn't mean we stopped moving our legs.  

Use of software or more generally, numerical computation is like doing something as blackbox approach. I believe that still the best way to understand and make physical interpretation of phenomena is via analytical approach.

While I think that tools such as Matlab are useful, they are more harmful to students and researchers who use it blindly as Ahmad so rightly puts it. These tools are empowering to people that not merely know how to use it but more importantly  recognize what are their limitations.

I think that overall we will dumb down. Just to give an example: During my studies we used Mathematica to simplify complicated equations. As it turned out at one point Mathematica did a reduction based on a wrong assumption. Up to that point we just used Mathematica blindly. Interestingly, the goal was to prove an equation by a famous computer scientist he proved in his PhD thesis. Only using the wrong assumption from Mathematica it was possible to get to the same result.

This doesn't mean that we should stop using these kinds of tools. But, everybody using these tools should learn what they are doing. Then, in case that something goes wrong they at least have a chance to figure out where the error is. As others have already mentioned: Tools like Matlab help us to put off cognitive load so that we can be more creative. In the end smart people will become smarter using these tools, and dumb people will become dumber.

Probably MATLAB is not the right tool, since it is a 16-digits accuracy floating point arithmetic machine, based on IEEE standards. I'd suggest working with symbolic algebra programs, where you have exact arithmetic and many other tools, that force you to learn mathematics (otherwise you cannot use them). The MATLAB toolboxes, although they are open to view them, they are so complex stacked together that you cannot modify them: once you changed something, the whole machine collapsed down.

"Are we becoming dumber or smarter using software like MATLAB to do our math?"

Well, on one hand we may become more efficient since we do not need to invest a long time in "tedious" mathematical developments.

But on the other hand, we lose fitness on our mathematical abilities and skills.

In my particular case I always start with pen and paper, this forces me to try to simplify the problem to start with instead of feeding my computer with a monster task I am afraid to accomplish.

But, fairly I shall acknowledge, I do use some mathematical tools (MatLab, SciLab, but also Maxima and other algebraic tools) for particular tasks to increase efficiency (weak point here can be argued, that I do not want to face myself these mini-monster problems).

I also check (I do not verify or validate my tools) whether the result is consistent with my expectations, and try to cross check and reverse the problem to see whether I go back to the original equations (easy example if I use Maxima to solve an integral for me, I may calculate with pen and paper the derivative of Maxima result to see whether I get back to my starting point, or at least I numerical approximate that integral e.g. in Excel to cross check results).

Now anyone who relies 100% on software like MATLAB (and others) may easily lose fitness in their math skills and abilities.

Further to that, it is not only that it might affect the final results of the research, but if the underlying theories are ignored may impede interpretation of the results.  Like the student who told me the kinetic energy of a expelled particle was negative... because solving the equation brought that result...

So I think we may benefit from technology, and we are more and more depending on technology (the easiest example is the low effort to do any school work by copy and paste from Wikipedia without even reading the Wikipedia article).  So I see a risk there, particularly for scholars...

But remember in the past all scientist could read and write in latin, now those skills are generally lost... are we dumber than our ancestors (of course I acknowledge I am dumber than most of them, and please do not take only Galileo, or Newton, etc... but a general scientist of the past)

As in many other occasions in history, time will tell...

I think it depends on the application for a particular software platform. For example, most lecturers throughout the World use Matlab to teach digital signal processing  (DSP) but this method has certain weaknesses. It is, nevertheless, an excellent tool for getting quick results.

I have used Orcad PSpice for 10 years as an alternative teaching tool to Matlab, even though the PSpice platform is used mainly for analogue and digital electronics. I have found that it produces excellent results, particularly if the students are weak in programming. In the main, the students said they prefer this method. Just recently, I have written a journal paper on using PSpice for teaching DSP but it was rejected because one of the three reviewers was quite negative about the technique! If anyone would like a copy of it then just email me.

MATLAB is useful tool, but often it can become enemy, for example in the cases when we do some mistakes and then spend hours, days or weeks to find what is the problem (and only then to solve it). I think that using of the matlab more than few hours in the day is only harmful to the brain.

Hello all,

It is good to hear all the comments for I am just joining the chorus. I must admit however for a little selfish reason. When I took my 3 sesmesters of Calculus the only help we had was beside ourselves was a TI 81 that we could program quadratic equation. The trouble was it did no good on test day. 

So the selfish reason is to make you students go through what I had to.

In jest,

Douglas 

Does MATLAB do our computations or does it do our math? 

I would argue that mostly it does our computations. So if ones aim is to be able to estimate computational outcomes (to develop confidence in the software outcome, say), then MATLAB will surely hurt us. If we wish to do serious math - develop new paradigms, prove theorems and such, MATLAB may hurt us - but maybe less. Symbolic computations can very often be verified (or, they should be verified) with numerical computations (with MATLAB!).  

Also, what would be an objective measure of our dumbness? What if I now find it hard to multiply 13 and 12 in my head, but can program MATLAB to solve boundary-value problems?

I think the theory and util is needed in parallel. But we must always begin with the theory why not even Excel.

We all know that there are "applied math" and "pure math". Traditionally applied mathematicians do a lot of calculations, whereas pure mathematicians much less. I think, by having "Computer algebra" packages push research to a more conceptual level, that it used to be. I find this a positive attitude.

On ther other hand, a simple user of these packages, is not expected to know what is behind the packages. It is just impossible. It is similar with all technological products. For example one does not need to be a mechanist in order to use a car!  

Dear @Ahmad, I find your statement very true : "our mathematical abilities are going down if we define mathematics through MATLAB only"! I was so surprised when I realised that some students did not know to solve some trivial equations without computer's help! Such practice and experience should be considered seriously in order to prevent possible consequences!

Further underpropping your remark above Costas, here's a quote by Max Tegmark (from his paper in "Universe or Multiverse?", edited by Bernard Carr, p.116) : 

"Modern mathematics papers are so abstract that the only numbers you will find in them are the page numbers"

Dear H Chris,

This is what I mean! Conceptual mathematics are about concepts and less numbers. An example of holistic-conceptual mathematics is Category Theory. See e.g. the Lawvere and Schanuel book: "Conceptual Mathematics: an introduction to Category Theory". I should explain the term "holistic" in characterising Category Theory. In Category Theory, by applying Yoneda's Lemma, every object in the Category is determined if we know either the ingoing arrows or the outgoing one. Thus there is no need for analysis of the objects. Thus category Theory is the best theory for living subjects!

Dear H Chris Ransford 

does this mean the future mathematics will be a combination of MATLAB functions? without low level contribution to for instance, Laragnge, Gaussians, Liemanns Eulers etc.

I think that we are becoming dumber using software like MATLAB to do our Math if done without learning the Math first.

Ahmad, there are 2 views on this, 1- maybe MATLAB is dumbing us down or 2- maybe MATLAB is freeing us from the nitty gritty of math to do what is really important in math.

The truth, as ever, is likely a blend of the two .

It is essential of course that students understand the essence of math, and a bit of nitty-gritty number-crunching is very important here.

Past these fundamentals, maybe trying to, say, integrate by hand complicated functions etc. does not contribute much to further mathematical enlightenment.

We have all read papers on, say, geometry, topology, etc. where it's all very conceptual and MATLAB would be hard put contributing anything there - papers by Penrose, Shin Tung Yau, Terence Tao, Andrew Wiles, et al. 

Beyond a certain stage, perhaps number-crunching becomes sterile and risks hogging our time and efforts, and keeps us from the conceptualization work so essential in bringing mathematics further forward ?....

Hello all,

there are two distinct phases in the process of converting a given problem into a computer program that can provide a solution:

(1) finding an algorithmic solution

(2) implementing this in a particular programming language.

It is argued that it is most important for the understanding of the students to perform the first step whereas the actual implementation in a programming language is of secondary importance for the learning of problem-solving techniques.

therefore there are many of researchers suggest to chose a well-structured teaching language that provides a clear and intuitive syntax and allows students to quickly express their algorithms.

In the below reference, they found that MATLAB is much better suited than C++ for this task but the best choice in terms of clarity and functionality of the language is provided by Python.