If one considers the historical evidences of thinkers contributing to the ideas that pertain to mathematics, the examples are aplenty. These include two basic categories of philosophers of mathematics: Western Philosophers and Eastern Philosophers.

Western Philosophers have some great names attributed to them such as Plato and Aristotle. Plato concentrated his studies on the mathematical objects, especially their ontological status. Aristotle, on the other hand, contributed to the field of logic of infinity.

It was the great mathematician Leibniz, who focused primarily on the relationship between logic and mathematics.

The study of philosophy of mathematics is made interesting due to the following aspects of mathematics:

o Mathematics is based upon countless number of abstract concepts.

o Wide application of mathematics: It governs many activities of our day-to-day life, besides its application in physics, chemistry and even biology!.

o Infinite: This notion is a peculiar one and has always aroused interest of many philosophers.

The relationship between mathematics and logic is one issue that has been a recurrent one in the philosophy of mathematics. In the 20th century, the philosophy of mathematics revolved around set theory, proof theory, formal logic and other such issues.

Around the break of the 20th century, there were several schools of thought that philosophers of mathematics held. At this time, three schools emerged, namely: intuitionism, logicism and formalism. In the beginning of the twentieth century, there was also an emergence of a fourth school of thought: predicativism. Any issue that would come up at that time, each school would aim to resolve that or claim the fact that mathematics is not as inevitable as opposed to those who believe mathematics to be “the most trusted knowledge”.

Logicism

It is the thesis that mathematics can be reduced to logic, thereby making it a constituent of logic. According to the logistics, the foundation of mathematics lies in logic and hence all the statements in mathematics are nothing but logical truths.

Simply put, this thesis suggests that mathematics is nothing but logic in disguise.

Intuitionism

This is attributed to the works of Brouwer. Intuitionism states that mathematics is an act of constructing. This involves mental constructions.

In this program of reforming the methodology of mathematics, it is believed that there exist no mathematical truths that have not been experienced.

Formalism

This program is attributed to the works of David Hilbert. According to Hilbert, the natural numbers can be thought of as symbols, and not as mental constructions, as opposed to the theory of the Intuitionists. These symbols are basic entities. And as far as higher mathematics is concerned; its statements are the strings of symbols, which have not been interpreted as yet.

Predicativism

Ordinarily, predicativism would not be considered as one of the primitive schools. This program is attributed to the works of Russell.

Now let us focus our attention towards the other contemporary schools of thought that have emerged in recent times.

Mathematical Realism

This program holds that mathematics is not invented by the humans, it is only discovered. For example, shapes like circles and triangles exist in the nature as real entities.

Empiricism

It is a form of realism. According to empiricism, mathematics can not be believed to be knowledge without experiencing (priory).

Mathematical facts can be discovered by empirical research. All the knowledge that is acquired is due to the observation that we make through our senses.

Formalism

The followers of this program are of the belief that mathematical statements can be viewed as the consequences of a number of manipulation rules applied upon the strings of numbers. There is another version to formalism: deductivism.

There have been many cases of mathematicians been intrigued and drawn to this subject of mathematical philosophy because of the sheer sense of beauty that they perceive in it.

]]>Is Mathematics Eternal?

Simple, complex, beautiful, elegant, ugly, explains all, success story, fundamental bedrock, etc. These are words and phrases often associated with mathematics, especially beauty and elegant. That in itself doesn’t make mathematics eternal.

I suspect that no matter what the laws, principles and relationships of physics were to turn out to be, there would be some sort of mathematics to cover it. However, much of our mathematics bears no relationship to our physics – inverse cube relationship for example.

Is Mathematics Eternal 2?

What is the status of mathematics? Is mathematics eternal?

Mathematics has no status outside of the human mind. So mathematics is only as eternal as the duration that human minds exist. Mathematics is an invention of the human mind (since I know of no other life form that makes use of mathematics in any abstract sort of way) to assist humans in dealing with the many (also invented) complexities of human society (like trade, commerce and economics). Mathematics provides practical applications like navigation and provides ordering and predictability in the natural world that rule the human roost. Mathematics is a not-thing since it has no physical properties and cannot be detected via any of your sensory apparatus. Of course if we’re in a Simulated (Virtual Reality) Universe then we totally exist as, and in, a mathematical construct.

Of course mathematics might also be the invention of extraterrestrial intelligences, so mathematics might persist eternally in the cosmos as long as there are intelligent life forms around to use and abuse their mathematical inventions.

Is Mathematics Invented or Discovered 1?

IMHO, mathematics is a not-thing, an abstract concept that’s the invention of the human mind. Mathematics has none of the properties that we associate with things. Things can be discovered; concepts are invented. One plus two equals three (1+2=3) is not a thing. Pi is not a thing. The quadratic equation is not a thing. Mathematical theorems are not things. Mathematics can not be detected with any of the five senses, or even with instrumentation that extends our sensory abilities beyond that which our sensory apparatus can come to terms with. Mathematics is a useful tool of course, though many possible mathematics that could be aren’t. We search around for and adopt the kind of mathematics that fits in with what we observe, with what is useful, and chuck what doesn’t fit in into the rubbish bin. So the gravitational force can be accounted for by an inverse square law, but not by an inverse cube law, so the inverse cube relationship is put into the rubbish bin. Then we wonder at the beauty and elegance of the inverse square law explaining the way the gravitational force operates over distance and forget about the non-beauty and non-elegance of the inverse cube law. As an aside, beauty and elegance are not legitimate scientific or even mathematical terms. You won’t find them in any scientific or mathematical dictionary no matter how often scientists and mathematicians use them as per many of the interviews here on “Closer to Truth”.

Is Mathematics Invented or Discovered 2?

The set of all possible mathematical equations is as close to infinity as makes no odds so it should not be surprising that a subset of those should by chance reflect what happens in the real world like the inverse square law for the propagation of electromagnetic radiation and gravity. That implies that mathematics is an invention and not a discovery. If there were really this wide wonderful world of a near infinite number of mathematical relationships awaiting discovery as something part and parcel and fundamental to the cosmos, then one wouldn’t expect that the vast majority ended up having no relevance to the cosmos at large and the laws, principles and relationships of physics that rule the roost.

]]>The Reality of Mathematics.

Mathematics is just a shorthand mental concept that simulates reality, or approximates reality or a possible reality or even an imaginary / impossible ‘reality’. Mathematics is NOT reality itself. You can mathematically manipulate the alleged extra dimensions in String Theory but that doesn’t mean of necessity that these extra dimensions actually exist.

Mathematics is a tool that at first approximation tries to reflect upon the nature of really real reality. Mathematics is not reality itself. Further, our mathematics are structured to reflect our version of reality based on our observations not of necessity what really happens. The perfect example is Quantum Mechanics. For example, we may not know, even cannot know even in principle, exactly where a particle is as well as at the same time where it is going with 100% precision. So we invent a form of probability mathematics like the Schrodinger Equation or the equation that governs the Heisenberg Uncertainty Principle. Those equations are for our edification but they don’t alter the really real reality fact that the particle has actual coordinates and is going from A to B. Probability in Quantum Mechanics, and the mathematical equations associated with it, are just reflections on the limits of the human observer and human instrumentation, not a reflection on Mother Nature’s really real reality. Our Quantum Mechanical equations are imposed approximations to really real reality much like Newton’s equation for gravitational attraction was really only in hindsight an approximation.

There can be multiple models of reality, each based on mathematics, but they can’t all be right. Cosmology is a case in point.

The phrase “but the mathematics works” means absolutely nothing. Just because mathematics predicts the possibility of some kind of structure and substance, or some law, relationship or principle that the Cosmos might have, does not of necessity make it so. A prime example where the mathematics worked but the Cosmos didn’t go along for the ride was the ad-hoc piling on those epicycles upon epicycles in order to explain the motion of the planets. It finally got so unwieldy that the baby was thrown out with the bathwater and a new baby conceived, that being that the Earth was just another planet and not at the center of life, the Universe and everything. Once it was postulated that the Earth went around the Sun, planetary motion fell into place – mathematically into place as well.

Take a more modern example. The mathematics works in String Theory, but to date String Theory remains a theorists’ theoretical dream (accent or emphasis on the word “dream”).

Probability theory is that branch of mathematics that interposes itself between the macro human and human comprehension and abilities and the micro world of quantum mechanics. That has way more to do with the macro than with the micro since the absolutes of the micro aren’t visible in the realm of the macro; they are beyond the realm of the macro to resolve through no fault by the way of human comprehension or abilities.

A prime example is that there is no probability in quantum mechanics, only probability introduced by the limitations of the conscious mind to get down and dirty to the level of detail required to eliminate the concept of probability from quantum mechanics.

Mathematics serves no purpose, useful or otherwise, outside of the context of the human mind (specifically) or outside of the intellectual conscious minds of other sentient species (in general), thus making allowances for E.T. and maybe the terrestrial great apes; whales and dolphins; and perhaps other advanced minds – perhaps elephants as well as some birds.

In the absence of any conscious minds, what use has the Universe for arithmetic, geometry, trigonometry, calculus, topology, statistics and the multi other branches of mathematics? Now 1 + 1 = 2 might be universally the case and logically true even in the absence of any conscious mind, or before any life form ever came to pass, but so what? That cuts no mustard with the Universe! There was nobody around to conceive of that or to make use of that or to equate the manipulation of numbers as a reflection of universal reality (or even non-reality*). There was no conscious or intellectual mind around to appreciate any mathematical utility or usefulness or beauty or elegance.

Mathematics in fact is not a reflection on or of reality, only that reality as observed or defined once having been filtered through sensory apparatus thus pondered over by the conscious mind. Reality as perceived in the mind is several transitional layers of processing removed from whatever pure external reality there happens to be. There’s even an additional layer if instrumentation is a middleman. So the conscious mind is thus limited in terms of its ability to come to terms with the full scope of really real reality.

Mathematics is the interface between humans and human comprehension, understanding, etc. of the Cosmos at large. Mathematics can tell you in actuality or theoretically the ‘what’ but never the ‘how’ or the ‘why’. For example, there’s Newton’s Law of Gravity, but even he realized that that equation just told you ‘what’, not ‘how’ or ‘why’.

The Non-Reality of Mathematics.

The following examples are some of what I term the non-realities of mathematics.

* Hyper-cubes are a nice abstract concept that mathematics / geometry can incorporate. However, while you might be able to play with real cubes, like dice, hyper-cubes will be forever beyond you.

* Stephen Hawking’s concept of negative time. Since IMHO time is just change and change is just motion, then negative time would have to be negative change and negative motion. That doesn’t make any sense at all. So while Hawking’s negative time might be useful in a mathematical sense, it has no bearing on our reality and can safely be ignored.

* Lots of quantum mechanical equations yielded up infinities so a sleight-of-hand concept called re-normalization was invented to deal with those cases involving infinities. That strikes me as dealing cards from under the table or otherwise known as a inserting a “fudge factor”. Does re-normalization represent really real reality?

* The mathematics of singularities inherent at the moment of the Big Bang or in Black Holes goes down the rabbit hole in that the laws, principles and relationships inherent in the physical sciences that are so otherwise adequately described mathematically now break down when trying to describe singularities and thus so does the accompanying mathematics that are involved as well. So what actually is the really real reality behind singularities?

* Mathematics are perfectly capable of dealing with alleged extra dimensions inherent in String Theory. However, that doesn’t make String Theory a reality, not does it make a half-dozen extra and hidden dimensions a reality.

* Mathematics is perfectly capable of dealing with an inverse cube law that has no correspondence with our physics. Just because a mathematical equation works doesn’t mean that there is a one-on-one correspondence to the real physical world.

* Mathematics are perfectly capable of dealing with zero, one and two dimensions yet these are just mental concepts that can’t actually be constructed and thus have no really real reality.

* Space-Time: Since space is just an immaterial mental concept (that imaginary container that actual physical stuff has to reside in) and since time is also just an immaterial mental concept (our way of coming to terms with change which is just motion – which is also an immaterial mental concept since motion itself isn’t composed of anything physical), then space-time has to be an immaterial mental concept. Neither space nor time nor space-time is actually composed of any material substance and the trilogy has no material 3-D structure. However, the mathematics involving the concept of space-time are a useful tool in describing reality, but not actually really real reality itself.

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