After reading this book, the one thing that seemed to stick
out the most (at least for me) was the question on how usefulness (or
uselessness) mathematics is. According to Hardy, something can be said to be
useful if “its development increases… the material well-being and comfort of
men, if it promotes happiness, using that word in a crude and commonplace way”
(116). There are many ways that one could interpret this. For example, someone
could say that because of mathematics, engineers are able to do their job and
build things. By doing their job, mathematics is contributing to the well being
and comfort of man because firstly, the engineers can get paid, which in turn
helps comfort the engineer, and secondly, whatever the engineer builds also
comforts the engineer’s client(s) since the engineer is building something for
them (whether it be a house, a building, a bridge, etc…). Another way this
could be interpreted is that by just simply sitting down and solving math
problems, this could promote happiness to a person that just likes doing this
type of thing (like a mathematician).
However, this also begs the question on how mathematics is
also useless. I remember there was a time in my Analysis class where my
professor told us that not everything in mathematics has a practical purpose.
There are some things in mathematics that (at least for now) only have a
purpose in mathematics. From this, I can see how some people may say that
mathematics (or at least, certain branches) are useless because they don’t benefit
anything except for the pleasure of mathematicians.
Anyway, according to Harding, he believes that ultimately,
mathematics is not useful (75). Personally, I think that mathematics is useful
since a lot of areas of mathematics does indeed supplement other areas of study
(like the sciences). Furthermore, just because something in mathematics doesn’t
serve a physical purpose now doesn’t mean that it wont always not serve a
physical purpose. At the bare minimum, those areas of mathematics can serve the
purpose of getting people to think and use their brain cells :)
You present some interesting points Phillip. Based on my understanding of the reading, Hardy brings into question the usefulness of math, or lack thereof, with the hope of illustrating purpose or meaning in his own life and those who dedicate themselves to mathematics. He suggests intellectual curiosity gets the ball rolling for such dedication to a subject matter but desire for reputation, position, power and money are also influential factors. Is a career not driven by the desire to benefit humanity respectable? I believe dedication to something, such as mathematics in this instance, does not have to provide an advantage to be useful or meaningful. This relates to the promotion of happiness as indicative of usefulness. So although set theory is not useful to the average person, it serves a purpose in the lives of others. This perhaps applies to the mathematical reality discussed by Hardy. Although the bulk of the population gets along fine without any in-depth exploration of this reality the importance of mathematics is not diminished. In my opinion, Hard was trying to justify his own lifestyle to himself. Quite clearly mathematics is an important subject to some and pointless to others and outlining its significance in the world, both real and applied, has very little worth in the measurement of a successful life or use of time.
ReplyDeleteMany of the concepts Hardy presented were concepts we have previously spoke/read about such as beauty in mathematics, a mathematician's prime, and the motives that drive intellectuals. I agree, Philip, that mathematics is more useful than Hardy portrays. He states that "a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself (89)." He also speaks of generality and depth being important factors in determining a serious theorem. To me it seems that there are plenty of mathematical theorems that lead to advances in mathematics and science and connect to other important concepts, enough to deem mathematics as useful. Basic mathematics also serves a purpose in the everyday lives of people who do not devote their time to studying it. As for the pure mathematics, I also agree that it may serve a great purpose and we should not be so quick to label it useless.
ReplyDeleteI agree with Philip and Rachel when it comes to seeing math more useful. I thought it was very ironic for Hardy to be presenting his veiws on mathematics the way he did, especially since the whole book is about mathematicians and mathematics. The way Harding talks about geometry and the example he gives (124-128), is like he really doresn't appreciate it. To me, his tone sounded a little demeaning to mathematics. He emphasizes the word "not" and uses words like: "plain" "imperfect" "distorted" and "nonsense". I just felt like he doesn't appreciate all the things mathematics proves and offers because it isn't something that the average human can use in a daily conversation or something that is a little more tangible.
ReplyDeleteHardy makes some interesting points on p. 135, which I think are relevant to this discussion. He discusses mathematicians who study "real" or "pure" mathematics and contrasts them with those who study applied mathematics. In doing so, he discusses "imaginary universes" and the wealth and richness of ideas which can come from not limiting oneself to those ideas which are applicable to the observable world we experience. I think this is an interesting idea, because lack of immediate apparent application is often motivation for ceasing development of mathematical ideas, perhaps prematurely.
ReplyDeleteI also think that Hardy's discussion of on p.117 is interesting and relevant, although I don't completely agree with his conclusions. Hardy states "the most 'useful' subjects are quite commonly just those which it is most useless for most of us to learn." I can understand his point that a select few experts may improve the state of being of the majority by inventing new technologies, and the "nuts and bolts" of these technologies need not be understood by the majority in order for them to be useful. However, I think a basic knowledge of elementary mathematics as a well as general background knowledge of commonly and frequently used technologies can be beneficial to everyone and help consumers avoid common pitfalls associated with certain technologies.
I listened to a good example, recently, of an instance when lack of basic mathematical knowledge caused embarrassment and public criticism of a well-known company. The link to the recording is here: http://www.funnyordie.com/videos/b4585f65f3/verizon-math-fail
I think Hardy's claim that mathematics is not useful came from the way he practiced mathematics. Mathematics was an enjoyable hobby for him that he liked for its own sake, so he did not see mathematics as a practical thing like others would.
ReplyDeleteI don't understand why Hardy would claim that mathematics is not useful, because it is pretty hard to find an area of mathematics that does not have a practical use for it. I think that Hardy is confusing motive and result.
If Hardy's claim that math is useless because it doesn't affect the lives of the average person is true, then there are very few disciplines that actually are useful. Ecologists, novelists, physicists, and artists might all be considered useless occupations according to Hardy since the average person doesn't care about the interactions of animals in the environment, or travel to a whole lot of art museums. Are these occupations actually useless? I certainly don't think so. What Hardy fails to recognize is that advances made in math affect many other areas of study as well. I once read a cartoon that said biology is essentially modified chemistry, chemistry is modified physics and physics is modified math. Therefore, everything in the sciences leads back to math. If this is true, Hardy is certainly wrong about the usefulness of math when one considers all the benefits we have reaped from developments made (and still being made) in the sciences.
ReplyDeleteHardy makes an interesting point distinguishing the different types of mathematics. Specifically he discusses the differences between pure and applied mathematics. Hardy defines as “There is the science of pure geometry, in which there are many geometries, projective geometry, Euclidean geometry, non-Euclidean geometry, and so forth. Each of these geometries is a model, a pattern of ideas, and is to be judged by the interest and beauty of its particular pattern.” (p 124.) Hardy says these are very different and are not pictures. Also , he says that the usefulness mathematics is not seen by people outside of the world of math. I believe what he really means is that the concept of pure mathematics is what people see outside of the mathematics field, see as not useful.
ReplyDeleteI agree with Philip and Rachel about mathematics having a useful place in our everyday lives. I also agree with chase regarding the tone of some of Hardy’s argument pertaining to certain areas of mathematics. I felt as if Hardy was being very critical of his own subject matter, isolating specific aspects of math and deeming them unworthy of passive interpretation. Although I found it strange and ironic for him to address geometry and other aspects of mathematics in the way that he did I can somewhat relate and seem to bring about some kind of understanding. I agree with Hardy in that geometry may help people and useful to certain things we do in this world, but as a whole we all don’t necessarily need to learn and understand it. Just because there are applications that we may use something for in our physical world does not truly determine whether or not studying its process is necessary or helpful for the average Joe. Maybe a better way we should approach teaching such subjects is by teaching the application that a certain mathematical concept is applied to, such as geometry in its uses to build bridges or find area. Teaching the basics I believe is necessary but going too far in-depth may be retroactive to actually getting people to remember what they have learned.
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