Foundations of Mathematics

When you’re quarantined with the flu you have a lot of time to sit and think (you also have a lot of time to ache, sleep, sweat, cough, and take in far too much daytime television), and this can be a good thing.  Of course, it can also be a bad thing.  If you’re like me, for instance, your thoughts turn to the foundation(s) of mathematics, which in turn lead you down inescapable rabbit holes and through familiar but frightening territory where you know you’ve been helplessly stranded several times before; you’re lost before you’ve even had a chance to chart out where you’d like to go.  Within a matter of seconds, your nausea is swelling and you’ve curled into the fetal position, muttering “Nothing makes any sense… nothing makes any sense…” all while Maury Povich blares ominously in the background.

I’d like to move things along on with a simple “Of course, that’s just me!” but, really, that’s not very honest.  The truth is that lots and lots of people have made themselves mad wondering about the foundations of mathematics.  (For those of us familiar with the difference between causation and correlation, its worth wondering whether this area attracts the attention of troubled minds or in fact troubles the minds of those who pursue it.)  This may strike you as odd since, like many people, you have some knowledge of mathematics already, and you’re not crazy.  Sure, maybe you’ve been afraid of math exams and formulas, but what is there to get so upset about?  2+2=4, and so what’s the big deal?

As it turns out, trying to build a firm foundation for mathematics is a tricky or subtle project.  When you try and distill it all down to a manageable list of starting terms and axioms or rules, not only are you disturbingly far removed from the (seeming) simplicity, applicability and big-picture-beauty of the subject, you are also faced with open-ended questions.  Are these axioms enough?  Does everything that should follow actually follow from this starting point?  Perhaps most agonizing is the question What does it all mean now?


“One cannot inquire into the foundations and nature of mathematics without delving into the question of the 34184169operations by which the mathematical activity of the mind is conducted,” warned L. Brouwer. “If one failed to take that into account, then one would be left studying only the language in which mathematics is represented rather than the essence of mathematics.”  In other words, successfully putting mathematics on a firm philosophical basis without concern for how we humans generate, appreciate, and value it reduces it to meaningless marks on paper.  For most (though not all), this indicates something is wrong, that we have gone too far down the rabbit hole.

So how should we — you, the new FOM student and I, the fairly-new FOM teacher — approach this subject?  For starters, we won’t resort to crying in front of our television sets.  No, we’ll have a textbook and class discussions to help tease out ideas and points of confusion, making such outbursts not only unnecessary but impossible as we’ll not have enough time for them.  Also, our journey won’t probe the foundations of mathematics too deeply; our starting point will not turn all of math into meaningless symbols.  We will take a few items for granted, and then, armed with them, carefully explore how mathematics works.  (If this whets your appetite, though, our department offers upper division courses perfectly suited for these interests!)

In other words, we will proceed with caution, keeping careful track of what we have assumed and what these assumptions allow us to conclude.  We won’t build a brand new house by starting with atoms, instead we’ll start with bricks and building supplies.  It will still be a good amount of work, of course, but its work that is long overdue.  After all, it may feel obvious that 2+2=4, but why is this true?  Exactly how was this house built?

To reground this post, a few more quotes are probably in order.  Keep in mind that, at least according to von Neumann, “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”  Words worth contemplating, especially after considering Russell’s claim that “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

If you’re feeling lost, I should point out one more quote that I find comforting.

“The lie detector test determined that was a lie.”  — Maury Povich

And, really, I should point out a bunch more quotes that will make you feel good, too.  At the end of a previous semester I asked all of my FOM students (pictured below) to share some advice with you, the then-future FOM students.  I’ve included their replies below and hope you find them useful.

Read them, think about them, and think about what you’d like to get out of this class. Lastly, go get a damn flu shot.




“Do your homework.  Work hard on it.  That’s how I learned things in this class.  Also, remember: there’s like 30 of you and one of him.  So stage a rebellion.”

“It may seem totally confusing at times, but its all part of the process, and you are not alone.  Just have faith in your abilities … Best of luck, and enjoy!”

“This is like your first real math class … Make sure you show up to class!”

“Math will make so much more sense after you take this class.  If you understand the definitions it makes it way easier to prove things.”

“This class will be mind bending and difficult, but you will leave knowing so much more about math that will help along your math career.  Study hard, the material of the class is difficult to grasp and it will take time, but you will feel greatly rewarded.”

“Do the work, especially the blog.  Definitely do the blog.  Yeah.  Do that.”

“Do me a favor.  Stop reading this letter for just a second.  Look at the people sitting around you.  You may have had class with some of them before, others may be completely new to you.  Now do yourself a favor and drop all preconceived notions you have of your classmates.  … Be willing to ask questions, and do not look down on others for doing the same.”

“The homework might take hours … keep up with the blogs … READ THE BOOK, it helps a lot. Have fun trying to prove stuff.”

“Enjoy being frustrated trying to prove stuff you know [or believe] to be true.”

“FOM is a matter of determination.  Your professor will be less of a teacher and more of a guide.  Most importantly, FOM is a process of knowledge, not knowledge itself.”

“Enjoy being uncomfortable, that’s what FOM is all about.  You will find yourself constantly challenged and that you like it and feel great when you have solved a problem. Also, write in your blogs and read your classmate’s blogs.  They really help!”

“FOM is not an easy subject, so don’t take it lightly.  ALWAYS read the chapter before class.  If you ever feel lost or fall behind, don’t hesitate to ask the TA or the professor.  Stay consistent with your studying and work together with others as much as possible.”

“Attend EVERY CLASS. … This class is really important if you want to pursue math, so pay attention. … You are awesome!”

“Don’t stress.  Make sure you take notes and ask lots of questions in class.  Make new friends, work together, and you will learn so much more than working alone.  The main point of this class is to reteach you how to think about math.  … Know definitions, use chalkboards to work out thoughts, talk about concepts, attend TA sessions, ask Casey to explain stuff, enjoy!”

“Brace yourself!  You should find this class to be challenging yet rewarding.  Take advantage of your resources and the help of your fellow classmates can provide.  But most of all, enjoy the complex beauty of the mathematical apparatus. …and remember, when math gets dirty, take a shower.”

“The work in this class can be difficult at times, and the ideas can be hard to follow, but work hard and all will end well.”

“You don’t actually know what you’re doing, but DON’T PANIC. Do the homework and set reminders on your phone to update your blog.  Ask questions (tricky ones, easy ones, anything).  Even if it doesn’t help you it may help someone else.  You will struggle to figure some things out on your own, so ask for help. … Seriously, its hard but don’t give up.  It’s fun and you’ll make it out okay.”

“My name is Inigoe Montoya.  You killed my father.  Prepare to die.  Also, be prepared to prove bunches of sh*t, whether or not you know it to already be true.  Oh, and do awesome projects!”

“You will be uncomfortable and find lots of things odd.  The most important thing is to be comfortable being uncomfortable.  Also, if you’re confused, go ask someone for help.  This is the way real math majors learn.  Have fun.”

“Good luck.  You are going to have your mind blown every single day.  The stuff you learn is going to be frustrating and make absolutely no sense at times, but your brain will explode so many times.  Read the book and keep an open mind because your current knowledge of math will be challenged.”

“This class will most likely be unlike any math class you have ever taken.  You will probably be confused the majority of the time, but by the end you will be able to look back and feel like you learned a lot.  Work on as much as you can in groups, it makes things easier and a lot more fun.  Don’t get overwhelmed and try not to worry.  If you work hard you will be fine.”

“It is helpful to read the book.  Also its good to prepare because if you have any questions before homework is due Casey will definitely help you out.  Anyways, don’t worry too much if you are ever lost.  Someone will help you.”

“There are a few things you should know about this class.  1. FOM is weird.  2. FOM is the art of clever trolling.  3.  FOM is no grammar.  4.  FOM is basically a drug that will blow your mind.  5.  FOM is thinking and exploring.  6.  It is work and reading.  7.  White boards and markers help a lot.  8.  FOM is everywhere.  9.  FOM is pushing definitions.  10.  FOM is proving.  This list can go on (but at least its countable).  You should believe in yourself and know that you can do this class.”

“… as long as you do your readings and participate in class you will learn a lot.  This class is also very fun (if you have a good class that does what they are supposed to do).  Write in your blog.  Also, it is really cool to go back at the end of the semester and look at everything you’ve learned.  The homework may seem difficult, but don’t get discouraged.  … You will actually have a good time and learn at the same time.”

“Welcome to a class that will blow your mind, if you let it. You should take the reading and homework seriously, and make friends!  Find people who are good at this and are nice enough to help you.  Take blogs seriously and do research into what interests you, it makes ALL the difference.”


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