(Update: I've posted a reply to some concerns here.)

Austrian economists hate math. That’s a broad brushstroke, but I stand by it.

And it's a shame. I understand the Austrian concerns about math and agree with most. But the concerns call for hesitation when applying math to economic problems. Yet, justified hesitation wrongly became abstention. I want to fight that tide.

This is not a post to bash Austrian economists and their lack of math. If you came here for a smackdown of Austrian economics, you will get no such pleasure from me. I'm not sure advanced math has been a net gain for economists.

Instead, this is an apology, in the religious sense, for math- pleading to young scholars interested in Austrian economics: Put away your concerns for a moment. Learn math. Math can be your friend. You will be a better (Austrian) economist for it.

Let me echo something I've heard Bryan Caplan say: Austrian economists are among the most interesting in the profession. My criticisms come from a love for the tradition and a wish to keep improving it. People who understand Mises/Hayek/Rothbard/Kirzner will push our understanding of the world in the coming decades. I'm sure of it. You, my Austrian friend, will help in that.

## From one friend to another

First, let me set up my street-cred within Austrian economics. I am no Trojan horse; I am truly a strong supporter of advancing Austrian economics. Whether other people will label me as an Austrian is up to them.

First, I am an Adam Smith Fellow at the Mercatus Center, which is **THE** heart of academic Austrian economics in the world. My job literally involves discussing Hayek and Mises with some of the best in the business. I believe they would vouch for me as a friend of Austrian economics.

Second, my research is clearly Austrian-inspired. One paper I'm working on examines the boom/bust Panic of 1907. The second looks at Hayek's theory of spontaneous order, specifically when that spontaneous order breaks down. I see my research as following in the footsteps of the greats.

Third, look at this blog. It's called EconPointOfView.com, a reference to Israel Kirzner's first book. The site's banner has Rothbard, Mises, and Kirzner- my way to show respect to those who I think are superb economists.

While none of what I've written makes me an expert on Austrian economics, I hope Austrians will be more willing to accept my point, coming from a friendly stance. I won't belittle those who choose to do a more literary form of economics. Do your thing. But along the way, pick up a math book.

## I agree with many of your criticisms.

I, as a good student of Austrian economics, accept the following:

- Economics is a value free science. It can never tell a person what he "should" do. My recommendation is not as an economic scientist but as a person.
- Pure economic theory derives from the action axiom.
- Math or statistics cannot be a source of pure theory.
- Statistical aggregates conceal most of the interesting information about the economic phenomena.
- Radical uncertainty makes point prediction impossible.
- Tacit knowledge, which can never be incorporated into formal theory as "information" or statistics, is the vital part of a functioning society.

I accept these points made by your favorite Austrians. These points invalidate how most economists use math. I still am urging you to study math.

Careful readers will have noticed something by now. I'm not urging Austrian economists to study mathematical economics. No. My central point is smaller. Study math, The Real Thing.

Math is pretty damn cool. Spend the time to learn some of it.

Obviously whether studying math is worth the effort or not, depends on each individual's opportunity cost. I know that. Instead, my goal here is to convince people that studying math has a higher expected benefit than they originally thought. Therefore, on the margin, some people will decide to study math.

## The Beauty of Proofs

While proof-based mathematics is by no means the only form of mathematics, it is the type I focus on here.

Mathematical proofs have aesthetic beauty, in the way music and art does. Once I started to understand proofs, I had a similar feeling to when I read Cicero in Latin; "I know I'm missing the subtleties, but Cicero is beautiful."

Since Austrian economists are mostly human, I urge learning math as an end in itself- like developing a palette for good food (which you definitely should do.) Math's pleasurable, in a sick way, and enhances your life, just like Bach.

## What is the purpose of a proof?

But you're more practical. You're an economist. You want Use. You want Reasons. Well, fine.

Proof-based mathematics attempts to create a solid connection between assumptions and conclusions, laying out every step between. That is not to say each connection is always clear.

I have written many proofs without truly understanding what I am doing. Still, by working to improve my proofs, I have developed a stronger understanding between the assumptions and conclusions. Little by little, I improved.

That's the main reason for Austrian economists to study math; **it helps train the mind in logical connections. **You say you love the logic of *Man, Economy, and State*. Studying proofs has helped me understand Rothbard's process.

Rothbard and Mises were smart enough to think about these topics in only words. I'm not. Math helped my thinking about the Austrian theory of utility, time-preference, and capital theory.

For example, I was discussing continuous preferences with a classmate. I was going through my basic Rothbardian explanation of the problems with continuous utility functions. Goods are never infinitesimally small, so a consumption set is discontinuous. Consumers either buy 1 apples or 2 apples, not pi apples. Boom, win for the Austrians.

Not quite. My classmate was able to explain to me that I also required an extra assumption: I need to assume the goods are finite. If there were infinite goods, continuous preferences is still workable.

It’s an obscure point, but I was simply wrong in my defense of Austrian economics. Knowing math helped me clarify my thoughts and communicate with a non-Austrian economist. Knowing a little about power sets, infinity, countability, and more was the only way that I could understand his (100% correct) point.

The math forced me to recognize another assumption I was implicitly making. I'm fine assuming finite goods, but now at least I know I am making it. Thank you math. I'm a better defender of Austrian economics because of math.

That doesn't show whether continuous preferences is a useful abstraction. Such decisions need judgement, something math cannot save our poor souls from needing. However, without an understanding of basic analysis, I would not have understood the connection between my assumptions and my conclusions. Heck, I couldn't even communicate with my classmate about this topic without proofs.

My example might seem small. But I'm not claiming that studying math will overturn your world. It will simply clarify points of detail. We learn on the margin. That's when math is most powerful; use it.

Is studying math the only way to connect assumptions and conclusions? Of course not. Economics teaches us that there are multiple means for every end. My point is that rigorous math is an effective way to pursue the end.

My dear Austrian friends: if you want to further your understanding of your own theories, drop the anti-math rhetoric. It’s unbecoming. Pick up a proof-based textbook and get studying. Challenge someone to learn the math with you. I'll be your partner if need be.

You spend time each day doing Sudoku or the crossword puzzle. Heck, you probably read Kant for fun. You enjoy challenging your mind. Why not learn math too? Plus it helps you communicate with other economists (which is worthy of a whole post).

So if you're someone who enjoys and wants to study Austrian economics, start reading your math too. Post comments or tweet at me (@BrianCAlbrecht) your thoughts along the journey. I'm going through the same journey with you.

## Starting The Journey

It takes time to learn how to write proofs; it does not come overnight. The long process continues for me.

I'm sure there is some expression about a single step and a thousand mile...

Here are my recommendations for a first step:

**Basic Proof Books-**

- Book of Proof by Richard Hammack (free online)
- How to Prove It by Daniel Velleman

**Analysis Books-**

- Understanding Analysis by Stephen Abbott
- Principles of Mathematical Analysis by Walter Rudin (the classic)

Over the coming weeks, I plan to write a similar apology for econometrics and computation. Both are tools that Austrians are rightly skeptical of, but wrongly reject outright. I'm partly a devil's advocate here, but still 100% sincere. (Or maybe it's just Stockholm Syndrome after my first semester in my PhD.)

Stop being so negative about anything outside of Rothbard and Mises. They weren't the only people to think about the world. Expand your reading list so that you can become better Austrian economists.

Coming from another Minnesotan, it should only be expected that I would advocate the importance of Math and the abilities it helps foster for the development of our dicipline. However, I believe there are aspects we are neglecting. Knowledge of Mathematics makes us erect our edifice of economic theory on the basis of outside demands. As in your example, we tend to think in terms of countability, compactness, continuity (etc) as basic starting points, which constrains our imagination. Maybe one who can look at the world without these lenses will make better sense (maybe some of the Austrians had better ideas than the heavily trained Americans)!

I really appreciate this post!

While I'd contend that there needs to be a discipline simply emphasizing the other aspects of rigorous econ-logic, and that math often becomes a grave distraction from the good stuff, I AM pursuing more mastery in mathematics and logic.

I thank you for the book-recommends!

A

I like the post, but recommending Rudin seems like a way to get someone to not enjoy math, especially if their primary interest in economics (or something other than math).

Great point. I was trying to recommend a specific type of math that I talk about in the post. I'd recommend the Abbott book first though. What would you recommend?

I haven't read the Abbott book, but it looks like a good book. For me the motivation should be at the forefront. Some good authors would be Thomas Korner and V.I. Arnold. An analysis book at the level of Rudin that I really enjoy is Stromberg's Introduction to Classical Real Analysis. It is out of print unfortunately.

Three cheers for math most certainly, Brian. But I found your post pretty confusing.

First of all, I really don't get your premise that "Austrian economists hate math." Hate! Huh? I don’t hate math nor do most of the Austrian economists I know, and I do know a few.

Then there is your example about continuity. I agree that we should reject the view that somehow utility cannot be continuous. IMHO that's wrong on so many levels! But let me stick with your post. As Jevons really makes clear we are typically talking about alternative rates of consumption, which can be any rational number. And I don't think most folks would object to modeling such infinite divisibility as continuous.

But even if we are talking about the number of units that might be consumed in a given time interval, I'm still not so sure I understand your discussion. Continuity as modeled in standard analysis in the usual non-Intuitionist math implies not merely an infinite number of points, but an uncountably infinite number. I could not tell from your account whether this came out in the discussion with your fellow student. You say "If there were infinite goods, continuous preferences is still workable," which is not a correct inference since a countably infinite set cannot be "continuous" in the usual sense. Maybe you meant what I was just saying? That even if the set is countably infinite, it’s an acceptable simplification to model it as continous? You do go on to refer to power sets and countability. So maybe it is your account just leaves off some details? Anyway, I found this part of your post a little confusing.

There’s something else. The assumption of continuity might be saved in a way that does not seem to have anything to do with being smart about math. When we speak of the utility of a good we are really talking about classes of goods such as “doughnuts.” Within the class there is usually an indefinite host of dimensions along which the goods vary and that matter for your utility. How big is the doughnut? How fresh? How much sugar was used in the dough? Was the clerk behind the counter polite? And so on. I don’t see how to model that with “discrete units.”

At the end of your post you say you’re going to post apologies for econometrics and computational methods. Same problem. Who among the Austrians is somehow opposed to either one? Dick Wagner, for example, is in pretty close communication with Rob Axtell at GMU. I guess I’m just wondering if you have accepted a model of “Austrian economist” that is *at best* outdated and more likely never legitimate. Remember, all those math-hating Austrian economists had to pass their courses and exams, so how much could they really math after all?

Roger,

Thanks for your thoughts. I'm sorry for not being clear in my continuity example. I didn't mean to get into a discussion of whether continuous preferences are a useful assumption. I believe it is for many situations. I hoping to quickly show (which I failed to be clear on) how my Rothbardian argument against continuous preferences didn't hold. My lack of clarity hurt my point.

As far as my articles on computation, I hope you'll be more sympathetic to my point. Most people I talk to interested in Austrian economics are not Dick Wagner. If Austrian equals Dick Wagner, who I've personally heard say he is "probably not an Austrian", as an Austrian, we can have very different discussions. For now, I'm using a very broad label and of course that has consequences.

To your final point- passing courses doesn't show any love for math. I know plenty of economics (even at Minnesota) who HATE HATE HATE macroeconomics, but still pass. Enjoyment of subject doesn't seem to be a necessary nor sufficient condition to making it through courses and exams. But maybe to us hate means different levels of emotion.

Thanks again for your thoughts. I posted a reply here with a few more points if you are interested.

I found this post confusing. I know there are a lot of untrained, self-described "Austrians" in the comments of countless economics websites who rail against math, but among trained economists, I think there's a lot more nuance in all of this. I consider myself a fellow-traveler with the Austrians. It' s not that I (and very likely other Austrians and fellow travelers) think math itself is bad, just that its use in economic theorizing is suspect, as Roger points out very well above.

I can't say I look forward to your posts on econometrics and computation methods if they are to include such sweeping and unjustified assumptions. In my reading of and discussions with Austrian economists, I've never felt discouraged from using econometrics in my work. In fact, nearly all my work is econometric in nature!

Your blog is very interesting and I look forward to reading it more.

Levi,

Thanks for you thoughts. I'm sorry my statements about Austrians, which I admitted were broad, distracted from my main point. That was not the goal of my post. I wanted to simply encourage "Austrians" who read this (who are mostly not you or Roger) to give math a chance.

I posted a reply on my blog if you are interest.

Thanks again.

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