# Why science must use math -- feedback welcome -- work in progress

Discussion in 'Science & Society' started by rpenner, Feb 15, 2012.

1. ### rpennerFully WiredValued Senior Member

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Why science must use math

The methodology of science has materially benefited humanity by increasing the knowledge of the universe by means of incremental progress. Year to year, in every area of study, what we actually know is non-decreasing. Consequent to the success of this methodology, professional scientists have been viewed as authorities in their individual areas of study and great weight is put on their opinions -- sometimes even when that opinion is not clearly based on their field of expertise.

Lots of people would like people to seek out their opinions, but comparatively few have the resources, inclination and desire to rise to prominence as a professional scientist in any particular field of study. Those that seek to partake of the authority of professional scientists without engaging in the actual methodology of scientists we call pseudo-scientists. One class of pseudo-scientist rests upon worthless "credentials" from a non-mainstream educational body, often for a price -- those are simple frauds. Another class of pseudo-scientist apes the form of scientific practice and may devote time to writing "papers" or seeking "publication" in a journal of ill-repute -- those we call cargo-cult pseudo-scientists after stories of South Pacfic islanders building shrines in the form of airport control towers speaking to non-existent cargo-delivering aircraft on non-functional "radios" made of woven plant matter. Yet another class of pseudo-scientist spends time trying to explain the behavior of phenomena that have not been documented to exist -- after Harriet Hall, we call those tooth-fairy psuedo-scientists after the example of someone who tried to work out the details of how a class of mythological beings obtained steady sources of local currency and for what purpose they sought to exchange currency for human deciduous teeth. A related behavior is science denialism -- the outright denial that science is progressively adding to human understanding, that it is of benefit or that it is reliable.

For both pseudo-scientists and denialists, a common trait which distinguishes this anti-science camp from professional scientists is antipathy and distrust of math. Here, I seek to explain that math is a necessary part of science and a large part of the reasons professional scientists are widely regarded as authorities.

I. What are the purposes and goals of science

Since the begining of recorded thought, mankind has recorded stories about what they thought Nature was created for -- birds sing because they were made for the beauty of their song or wood was made to float or goats were made to be tasty when goat parts, garlic and chili powder are placed on hot metal. Back in the 17th century, Bacon rejected that egocentric tradition of teleological story telling in favor of proceeding empirically to accumulate and organize data and work out inferences from that data. Further such inferences can be tested by further observation to see that are reliable.

a. To gain knowledge of phenomena which occur in the Universe
Gathering data is a goal unto itself.
b. To organize and unify knowledge of phenomena which occur in the Universe
A hypothesis which explains many events reliably is easier to teach and learn than a great many events unconnected by unifying principles.
c. To gain influence for humanity over the operation of the Universe
"Knowledge is power" - Bacon
If we have communicable, reliable and precise descriptions of the behavior of nature, then we can predict in detail what happens when we make choices. And if we can predict what the outcome of various choices are, then we can decide objectively between the choices. Then, of these choices, we can choose the best alternative at the cheapest cost. What more power could one aspire to?

II. What is the methodology of science

Observation, Model-making, Testing and Unification are important parts of scientific methodology.

a. Observe a phenomena
Scientists don't just observe. They observe repeatedly and in detail. They gather data that is obviously relevant and data that almost certainly is not relevant. They gather data when the object of their study is present and also when it is absent, all the better to compare with.

b. Make a hypothetical model predicting aspects of the phenomena
From the great mountain of data, a trend, a cluster of points, a distribution or a

c. Test and reject models that don't reliable predict the phenomena

d. Where possible, unify models to explain multiple phenomena

III. Is a fragile model of more or less use to science than a flexible and ambiguous model?

IV. Why are mathematical models fragile?

V. Is this fragility generally seen in other aspects of mathmatics?

VI. Is this fragility generally seen in Literature or Philosophy?

VII. What are the results of math-based science?

2. ### ughaibuRegistered Senior Member

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No mention of the Field/Balaguer nominalisation project?

3. ### riverValued Senior Member

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4,194
mathematics is important , no doubt

but its not the paradigm of thought

4. ### arfa branecall me arfValued Senior Member

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Computational languages and digital computers are one result. With which we can model physical systems like atomic weapons (not good) and also supernovae (better).

But also genetics, proteomics, complexity theory, quantum mechanics, ... Lots of stuff.

Physical theories are analogous to algorithms. A mathematical theory that describes a physical law is like a program that runs "correctly". Computers are like a proof of something too, by the fact they exist.

What is this proof? That logic is "real", or that numbers are? Is there really any difference or is it just down to how a program (and the computer it runs on) interpret strings of binary? Or does the existence of modern computers prove something else; (that humans are "clever" maybe)?

Not sure. I think I understand what a "robust" model means, though.

I can't quite envisage what's fragile about a number or a function that takes numbers to numbers. I suppose the halting problem could represent a certain fragility, which leads to incompleteness theories. Perhaps it's because theories are axiomatic, so the fragility is about choosing the axioms. That is, axiomatic logic is based on a frail choice of "initial" or "a priori" data.

Or maybe I don't know what I'm talking about.

Last edited: Feb 15, 2012
5. ### riverValued Senior Member

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4,194
because they are based on what is known

6. ### leopoldi miss my coco.Valued Senior Member

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17,015
i do not consider math models fragile.
they are either right or they are wrong, "close" is not right.
a right math model is about as solid as you can get without the actual reality.
-my opinion.

7. ### ughaibuRegistered Senior Member

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Your overall thesis needs to be clearly stated. As it stands, "science must use math" can be interpreted as having a complete meaning, such as, 'there is no science without maths', to a minimal meaning on the lines of, 'within science there is at least one statement which can be expressed mathematically'.
It's difficult to see how meaningful science can avoid predicting observations, and as predictions have, at least, the form of if, then statements, such predictions can be expressed mathematically, which might make your thesis trivially true and thus, uninteresting. However, this doesn't entail that the relevant statements of prediction cannot also be expressed in natural language. So you'll need to be clear about how you draw the line between what is and what isn't mathematics. For example, the expression of measurements in numbers, does that amount to mathematics?
This statement is rather dubious. The high profile denialist positions are against evolution, free will and global warming, in none of these cases is the question of mathematical au fait pertinent. Notice also that denialism isn't necessarily anti-science, for example, it's more or less entailed by scientism. It's also unclear that all which is classed as pseudo-science is anti-science, so these matters, too, will need to be clearly addressed.

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9. ### arfa branecall me arfValued Senior Member

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--http://bjps.oxfordjournals.org/content/42/2/147.abstract

--http://en.wikipedia.org/wiki/Stochastic

So is the idea of mathematical fragility tied to the idea of needing a sufficiently large sample, if the universe is statistical?

Last edited: Feb 15, 2012
10. ### EmilValued Senior Member

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2,789
Mathematics is very important in certain areas of applied sciences.
But not in the models I have seen displayed on this site, but in mathematical modeling of dynamic systems.
These models can be simple but can reach very high complexity
Worth a look at these links.

Mathematical Model of Physical System , Modeling and Simulation of Dynamical Systems

For those interested, I suggest to browse the many pages on the following link : Modeling and Simulating Dynamic Systems with Matlab

11. ### rpennerFully WiredValued Senior Member

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Not yet, and I don't expect to. At best, this project has advanced the case that: "if a nominalistic claim follows from a purely nominalistic theory extended by a purely mathematical theory, then it follows from the nominalistic theory alone." (Colin Cheyne, Knowledge, cause, and abstract objects: causal objections to Platonism, p. 166)

But the examples so far have been trivial, and don't seriously address day-to-day quantum phenomena like Pauli exclusion. Regardless of whether in some abstract sense complex numbers exists, quantum mechanics requires the math of complex (or matrix) multiplication to simply model the complex phases seen in entanglement and that separate quantum mechanics from probabilistic Newtonian mechanics.

The most damning part of this nominalization project is that it does not advance mankind. It follows afterward -- like someone color coding all the numbers in the exercises in a 10-year-old's math textbook by the remainders modulo six. It's a type of intellectual exercise that might appeal to your aesthetics, but doesn't help you learn integral calculus any better.

It's unclear if there every can be a test of Platonism or anti-Platonism, but it hardly fits with the pragmatism of empiricism to attempt constrain a priori the universe to just models which are aesthetically pleasing to one's preconceptions. If the universe doesn't care how we think about it, then we must not impede ourselves in our study of the phenomena of the universe. I do not see nominalism as improving how humanity thinks about the universe. When Balaguer spends three paragraphs trying to make "Mark's boat is 50 feet long" acceptable to nominalism, I think that boat has sailed.
http://findarticles.com/p/articles/mi_m2346/is_n418_v105/ai_18262771/

Physics is hard. Math skills make it easier. Worrying about straying from the path of nominalism doesn't seem to make it easier or (importantly) any more useful.

It is however a model of precision in communication and utility.

Maths are fragile in the sense that if ever you show that $x \neq x$ or $1 = 0$ or $0 * x = 1$, this one contradiction causes the whole utility of mathematics and logic to crumble into nonsense.
http://xkcd.com/704/
http://en.wikipedia.org/wiki/Principle_of_explosion

It is "there can be no advancement of science without math." I'm talking about "doing science" -- not the application of rules of thumb previously obtained.
The high profile denialists are all pretty dodgy at math.

http://scienceblogs.com/deltoid/2011/11/rosegate_rose_hides_the_inclin.php
http://scienceblogs.com/goodmath/2008/08/why_is_randomness_informative.php
http://scienceblogs.com/goodmath/2007/04/george_shollenberger_returns_t_1.php

12. ### RichW9090EvolutionistRegistered Senior Member

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No, that is not true. What we actually know is not staying the same year over year - which your phrasing of this statement allows. What you should have said is "Year after year, what we actually know is increasing".

13. ### rpennerFully WiredValued Senior Member

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If the area of study is small enough, there are times when we don't add to our knowledge of the subject. Example: Mankinds knowledge of the backside of the moon was staying constant from 1800 to 1850.

14. ### RichW9090EvolutionistRegistered Senior Member

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True enough, but that statement as made was about "what we know" as a whole, not in reference to any small area of study,

rich

15. ### rpennerFully WiredValued Senior Member

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As the author of that sentence, I believe I am the better judge of the author's intent than you.

16. ### wellwisherValued Senior Member

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Math is important but it is secondary to conceptual modelling. For example, say I wanted to use astrology to explain the astronomy of planet orbits. The ancients plotted the path of the planets and put this in charts. I could mathematically model this to gain math support. The math may be perfect, but the theory would still fall short, since conceptually there are assumption problems. The math is like a horse that you lead.

You need to be careful about math razzle dazzle being used to cover up bad premises which need to be updated. Often the math is very advanced to where we get bogged down in the math and fail to look at the assumptions behind the math.

In my experience, there are also special effects one can create with math which allow one to even model assumptions which are not part of physical reality. For example, there is an abstract drawing called the stairway to nowhere (below)/

It shows a unique stairway, that can exist on paper, but not in the real world since ti goes up or down forever. Say this drawing was a data plot. We can equations that allow us to express this data. Now we can use that math to predict things that can not exist in reality. I trust math will be right, but still like to look at assumptions to make sure the math is not supporting an illusion.

Consider a phenomena in science that has two competing theories. Both can be supported by math, so they are professional looking. If it turns out, only one was correct, math supported an illusion that looked real.

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18. ### impaJahRegistered Senior Member

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Excellent post wellwisher!!

I don't think anyone except for legitimately crazy people would deny the amazing utility of mathematics. The particular backlash I think you're talking about, rpenner, is when math is used more as a wall rather than an aid to communication (on these forums). From what I can tell I don't think you do this, but I also don't think you could deny that there are some people that would rather confound others with a page of formulas, not caring whether the person understood or not, in an attempt to "win" the discussion.

It's the same type of person that refuses to change their vocabulary for their audience. What's the point? It's all just self indulgent "look what I can do" behavior.

And deep down everyone must know it's bullshit... if only because mathematics is just a form of short hand of the english language with agreed upon definitions of things. Therefore it can always be expanded and it's inner logic available to the mathematically uninitiated.

It could also be that the people who do this have only learned the equations by rote and don't themselves recognize or understand the underlying logic. In all likelihood this is the case, because if it weren't I can't imagine how a person would not enjoy contemplating the underlying logic over slandering and attacking a person trying to do just that.

But I guess this is what happens when a person's own self worth is invested in something as flimsy and untenable as their ability to crunch numbers and scrawl formulas. They can't take down that wall - who would they be without it? Who wants a level playing field in a competition?

19. ### TrooperSecular SanityValued Senior Member

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I would think that a fragile model is more useful to society because we learn from our mistakes, but this process of weeding out is taking much longer now, as experiments are more difficult, time consuming, and more expensive.

Are we at a period in history where we are limited to aesthetic models? If so, then we are entering into another golden age of crankdom. There was a small window period when advances in science became increasingly dependant on expensive experimentation, pushing out the dabblers and enabling experts to become more exclusive, a boundary once drawn between the experts and the foolishness of the inexpert masses. A time that assisted in the purging of pseudoscience, but the internet is allowing a new era of undisciplined knowledge back into society. It is a trade off with innovation. We have to expect of certain amount crankery whenever there is rapid increase in communication and consolidation of information.

What is the etymology of the word “crank” anyway? Is it crooked? It is unfortunate that both "cranky" and "crotchety" took on the meaning of crabbiness instead of eccentric opinions.

Has anyone read “Some Cranks and Their Crotchets” by John Fiske?

"Crotchet" and "crochet", both related to “hook” and “crook” have taken on different meanings, but it is somewhat ironic how Margaret Wertheim is dabbling in both.

Which do you prefer, RP, and why, pure mathematics or applied mathematics?

Last edited: Feb 19, 2012
20. ### HectorDecimalRegistered Senior Member

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438

I'm very glad you didn't discredit progenitors of a particular style of your description of true scientists because they reject academic job placement and prefer to work indepenendly as a contractor or consultant. I occasionally get the quaestion about why do I use such complicated math for what are sometiimes simple problems. I generally answer by explaining that within every simplicity, transparent to the non-scientist, is a host of complexities that must be refined not only through some mathematically reductive process, but, in line with your description at large, further refined by repetition.

We take a model, then enough varieties and repetetive varieties to create a standard, then test the standard against deviations, extract from that a standard deviation and repeat the process. This is why science, performed on a professional level, is so costly. These processes all take a lot of time.

I think the key distinguishing element between a true scientist and a pseudo-scientist is not his personal idiosynchracies or adapted methodology, but his willingness to back away from a project if a potential client wants him or her to cut corners. It's just good business. It's far better to disappoint someone honestly and tighten the belt, than to mislead a client with an oversealous lie. The latter will almost always come back to haunt the contractor, scientific or otherwise. I'd starve before telling someone I can do a $10k project for$2k.

Nice thread. I hope many can learn from it.