# Mathematics and the real world

• nightflyer
In summary, mathematics is considered to be an effective language for describing the empirical world due to its ability to construct objective models that can be examined and reach the same conclusions. It is also believed that the brains that conduct and observe empirical experiments are naturally the same type of brains that develop mathematics, and that the empirical world is somehow bound to function based on mathematical principles on a metaphysical level. However, there is not just one "mathematics" but many different "mathematical theories" that act as templates for applying mathematics to real-world problems. The success of mathematics in predicting and explaining the natural world is also attributed to the rules of inference and the assumption that these rules hold in the world. There
nightflyer
Why is mathematics - a language developed through reasoning - so effective when it comes to describing the empirical world? Could either of the following have anything to do with this phenomenon:

- the brains that conduct and observe the empirical experiments are naturally the same type of brains that develop the mathematics: human.

- the empirical world is somehow bound, on a metaphysical plane, to function based on mathematical principles.

I would also like to raise another question, which I have touched upon in previous posts, but would like to discuss in more general terms. Is mathematics a good tool when it comes to describing biological systems, or are they too complex to fit into the models?

Any thoughts, links or other references are much appreciated!

All the best,
Anders.

I think the thing is that mathematics is the easiest language / tool with which to construct objective models, i.e. models that two people, even if they disagree, can examine and reach the same conclusions. The tradition of science has frequently found that constructing a mathematical model of a real-world phenomenon is a powerful tool to accompany experimentation and that these mathematical models are also usually the easiest way to clearly communicate concepts.

Mathematics may potentially explain biological systems, the question is whether or not the mathematics can develop that mathematics =]

I once read a response to the idea that science fiction has predicted so much of what has actually "come true"- science fiction has predicted many things, most of which have not come true. But if you predict enough things some of them are bound to come true!

The key point is that there is not just one "mathematics"- there are many different "mathematical theories" (group theory, linear algebra, Euclidean geometry, etc. are all different "mathematical theories"). Mathematical theories are "templates". At the basis of any mathematical theory are "undefined terms" and axioms using those undefined terms. To apply mathematics to a "real" problem, we look for a "template" (a mathematical theory) that "fits". That is, we assign meanings, from the problem, to the undefined terms. If with those values, we can show that the axioms are true, then we know that all theorems derived from the axioms and all "methods" derived from those axioms will work for this problem. Of course, the best we can hope for is that we can show that the axioms are approximately true. Since real problems always involve measurement, which is approximate, we cannot expect them to be exactly true.

HallsofIvy said:
...there is not just one "mathematics"- there are many different "mathematical theories" (group theory, linear algebra, Euclidean geometry, etc. are all different "mathematical theories"). Mathematical theories are "templates"

That's why I believe that mathematics, taken as a whole, may be a larger (or perhaps more varied) invented structure than discovered reality itself. There seem to be parts of mathematics which don't correspond to parts of reality, spare mathematics, as it were, from a physicists point of view. Perhaps Hamilton's Quaternions are an example? But then what look like spare parts suddenly turn out to be very useful -- Tensor analysis for General Relativity for example. One never can tell!

As to the original question:
nightflyer said:
: Why is mathematics - a language developed through reasoning - so effective when it comes to describing the empirical world?
, I've set out my ten cents worth in another concurrent thread: (https://www.physicsforums.com/showthread.php?p=1611545)

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Of course, as an addendum to Halls' post, with a "mathematical template", we should ALSO understand that set of rules of inference (i.e type of logic) that allow us from the axioms and undefined terms to deduce what else might belong in the template.

Furthermore, when modelling, we must assume that the world, or the world facet ALSO abides by those rules of "deduction".

arildno said:
Furthermore, when modelling, we must assume that the world, or the world facet ALSO abides by those rules of "deduction".

That's not entirely true. A great deal of scientific progress has been made with completely incorrect scientific models - corpuscular optics, phlogiston chemistry, the Rutherford atom. Inaccuracy in a model certainly limits its usefulness but by no means does it make the model useless.

For example, QM and GR are very far from being integrated; at least one of them must have major errors or inadequacies. But that doesn't prevent them from being essential to science.

CaptainQuasar said:
That's not entirely true. A great deal of scientific progress has been made with completely incorrect scientific models - corpuscular optics, phlogiston chemistry, the Rutherford atom. Inaccuracy in a model certainly limits its usefulness but by no means does it make the model useless.

For example, QM and GR are very far from being integrated; at least one of them must have major errors or inadequacies. But that doesn't prevent them from being essential to science.

Hmm..you misunderstood me, possibly because my bad phrasing:

Any prediction made from mathematical modelling "should" be understood to have the form :

"INSOFAR as our rules of inference holds in the world, and the structures in it has been correctly represented in our model, THEN it must hold that we will see...blah-blah".

That is what I meant by "must assume".

Ah, I see. Yes, that seems like it's an assumption behind employing models in science. Though it definitely seems to me that when theorists are having fun they just go nuts and may not be too concerned with that.

CaptainQuasar said:
Ah, I see. Yes, that seems like it's an assumption behind employing models in science. Though it definitely seems to me that when theorists are having fun they just go nuts and may not be too concerned with that.

Not too concerned??

That's their main concern!

For example, if the various models within string theory hadn't, say, been able to imply general relativity, no one would have bothered with them.

And if some such model predicted something utterly wrong, for fun's sake that whales tend to fly, particularly on midsummer eve over the Scottish highlands, then that theory would end up in the garbage bin.

What theorists play with are alternate models, trying to deduce divergent predictions of those models, disputes that hopefully can be settled by some experiment.

However, they need to "play" quite a lot with their models as mere mathematical beasts if they are to master them in such a way that they can yield such predictions.

you've got to get to know the beast you're riding on, and then you first must de-focus on where you want it to go.

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You're saying that theorists never get overly entranced with the intricacies of the model they're building and veer off down a path that an experimentalist would never look twice at? You're a theorist, aren't you?

## 1. What is the importance of mathematics in the real world?

Mathematics is essential in understanding and describing the world around us. It helps us make sense of complex data and make accurate predictions about real-world phenomena. It is also the foundation for many fields such as science, engineering, and economics.

## 2. How is mathematics used in everyday life?

Mathematics is used in a variety of everyday activities, from calculating grocery bills and managing personal finances to measuring ingredients for cooking and telling time. It is also used in problem-solving and critical thinking, helping us make informed decisions and solve real-world problems.

## 3. What are some real-world applications of mathematics?

Mathematics has numerous applications in the real world, including in fields such as physics, biology, medicine, and finance. It is used to model and understand natural phenomena, develop new technologies, and make accurate predictions about the behavior of complex systems.

## 4. How does mathematics contribute to technological advancements?

Mathematics plays a crucial role in technological advancements, providing the theoretical and analytical framework for developing new technologies. It is used in fields such as computer science, cryptography, and data analysis to design and optimize systems and algorithms.

## 5. Is mathematics the same in the real world as it is in theory?

While the principles and concepts of mathematics are the same in theory and in the real world, their application can differ. In the real world, factors such as measurement errors, environmental factors, and human error can affect the accuracy of mathematical models and calculations. Therefore, it is important to constantly test and refine mathematical theories and models to ensure their applicability in the real world.

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