What then is Mathematics?

Kurt Gödel’s Proof of the Existence of God

Anyone who thinks there is a simple answer to that question will be seriously disappointed. What have famous mathematicians said about it?

Mathematics is the art of giving the same name to different things (Henri Poincaré)

Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true (Bertrand Russell)

Is mathematics something ‘out there’ waiting to be discovered, or is it a product of the human mind? Do we discover mathematical laws and relationships in nature, or do we impose mathematical descriptions on nature as a way of making sense of it, and harnessing it? Do we even know what number and arithmetic are? Yes, we all know what five loaves and three fishes are because those numbers have referents, but what of the number ‘three’ apart from its being a word or a symbol – does it exist as an immaterial entity? If so, where?

Questions such as these have interested philosophers and mathematicians for centuries. There are at least three fundamental questions to be addressed, on ontology, epistemology, and truth:

  • What is the nature of mathematical objects?
  • How do we obtain knowledge of them?
  • How do we account for certitude in mathematics?

We could add another question on effectiveness:

  • How do we account for the utility of mathematics in physics?

The various views on these questions correspond to the two camps of the medieval philosophers, the Realists and the Nominalists (or Anti-Realists), though Realism goes back much further to ancient Greece. Essentially, Realists believe that abstract entities or universals exist in their own right independently of the mind that thinks them, whereas Nominalists deny the extra-mental reality of universals and abstract ideas.

Continue reading ‘What then is Mathematics?’

The Du Sautoy Code

Professor Marcus du Sautoy

Marcus du Sautoy, Simonyi Professor for the Public Understanding of Science (having succeeded Richard Dawkins in the Chair), is currently presenting a series of TV programmes about mathematics and nature entitled ‘The Code’. Viewers could be forgiven for believing that what he is presenting is a mainstream view of mathematics rather than peddling his own peculiar brand of atheistic metaphysics. Since no appropriate caveats have been employed by the BBC, we feel it necessary to make a few of our own.

Firstly, Du Sautoy’s view that, as the Pythagoreans expressed it, ‘Number is everything’ is of very ancient pedigree; but, nothwithstanding, it is undemonstrable (which should be anathema to a mathematician) and a faith-based religious concept. Secondly, philosophers of mathematics and informed students of mathematics know that there is, to date, no satisfactory understanding of the relationship, if any, between mathematics and reality; to suggest that there is a relationship, and what such a relationship might be, is an act of faith. And thirdly, it is very unfortunate for scientists to be working with mathematics as though mathematics itself is the original reality to which the physical world ‘must’ conform through such things as ‘laws’; science has been hideously corrupted in the last 80 years because of this.

Some Christians might be heartened to see and hear Du Sautoy suggesting that numbers are at the root of all reality, that this is in some way all grist to the mill of Intelligent Design. Not so fast: Du Sautoy is an avowed atheist (who not very wittily gives his religion as ‘Arsenal’) who by his own admission is trying a more ‘softly softly’ approach than Richard ‘The Rottweiler’ Dawkins (whom all can see is a bigoted fanatic) and is not appealing to design, or even apparent design, but to some mysterious entity he calls ‘The Code’. A code at the very least implies information content, but The Code (as a proper noun and with the definite article) suggests something unique and powerful. Thus Du Sautoy:

…underlying everything that surrounds us, from the natural world to the cities we live in, there is a hidden code that explains why things look and behave they way they do.

[This hidden code (‘The Code’)] has the power to unlock the laws that govern the universe.

The Code is the truth of the universe, and its numbers dictate the way the world must be.

So, this hidden code, this entity that Du Sautoy calls ‘The Code’, has total and complete explanatory power, is identical to Absolute Truth, can lead us into All Truth, and is completely deterministic. This is unquestionably a religious position. And it is none other than the old heresy of Pythagoras, the pagan Greek philosopher, re-worked by gnostics, Kabbalists, Freemasons, Rosicrucians, Illuminists, and now, it appears, New Atheists. What a wheeze if they can pull this one off!

Continue reading ‘The Du Sautoy Code’

Copernicus and the Lutherans

Luther's room where he engaged in 'table talk' over meals with his students

It never ceases to amaze how scientific myths manage to get recycled ad nauseam. We have all heard the absurd myth that people once believed the earth was flat, but then (since characters are involved) there is the more mischievous myth that the Protestant Reformers, Luther and Calvin, vehemently opposed Copernicus. In truth, you will find nothing in their writings, their letters, their sermons or any other productions where Copernicus is even mentioned. But, amazingly, in textbooks you will find that “Luther attacked Copernicus” (Berman and Evans, Exploring the Cosmos).

I recently read a comment stating that Martin Luther called Copernicus “the fool who will turn the whole science of astronomy upside down” and that Copernicus found it difficult to get his work published by the University at Wittenberg, and that his supporters found it difficult to get or retain jobs there.

You will search in vain within the large corpus of Luther’s works to find the abovementioned quote, or anything like it. If anyone has the candour to give a reference it will finally be traced back to an entry in the Tischreden or ‘table talk’, which contains a lot of highly entertaining but doubtless embellished and spurious material that purports to be things Luther said at the dinner table. These sayings were written down by dinner guests decades later and are practically worthless as a historical source.

Continue reading ‘Copernicus and the Lutherans’

Galileo’s vain ambition

Galileo’s Dialogue concerning the Two Chief World Systems (1632), which argues for the superiority of the heliocentric Copernican over the geocentric Ptolemaic system, is a classic straw man argument. By 1632, astronomers had all but abandoned the Ptolemaic system, which had become untenable in the light of the evidence from telescopes for more than 20 years, which showed, for example, that Mercury and Venus orbited the sun. By that late stage, the majority of astronomers had adopted the geo-heliocentric model, such as that of Tycho Brahe, which had all the planets (other than earth) orbiting the sun, and the sun and the moon in orbit around the earth. Adapted to include a rotating earth, it was indistinguishable from the Copernican system on the basis of empirical observations of our ‘solar system’ from the vantage point of a terrestrial observer.

The Tychonic system was still highly regarded until the end of the seventeenth century when Newton’s discovery of principles of gravitation, and their application to the solar system, caused it to be abandoned as an explanation of reality, though it continues to this day as an elegant method for generating the terrestrial observer’s view of the motions of heavenly bodies in our solar system, for example in planetarium projectors.

So, Galileo was incapable of making a case in favour of Copernican heliocentrism over against the majority view of Tychonic geo-heliocentrism based on observation of the sun, moon and planets, and he could not do so based on physical principles as he rejected the idea of gravitation beyond the earth’s atmosphere and had never bothered to read up on Kepler’s laws of planetary motion, which had been sitting on his bookshelf for over 20 years. He also believed that no practicable terrestrial experiments could establish whether the earth was in motion, for he says in his foreword “all experiments practicable upon the earth are insufficient measures for proving its mobility, since they are indifferently adaptable to an earth in motion or at rest.” Thus he deliberately contrived to make his case using the artifice of a straw man argument, with not even a mention of the then chiefest of the world systems, the Tychonic model.

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Hawking’s Grand Delusion (Part III)

The Fool, in his own words

[Read Part I and Part II for background]

Stephen Hawking is doubtless a very intelligent man, but in his most recent book The Grand Design (surely a title that is supposed to be ironic) he has shown that even the most intelligent of scientists can write like a fool, and this monograph will become a classic for that very reason. He followed up his inanities in an interview on Larry King Live on September 10, 2010. It is now evident to all (if anyone was hitherto in any doubt) that Hawking’s brilliance is in a very narrow field indeed, apart from which he gropes and stumbles like a drunken man. Early in his book he announces

Philosophy is dead. It has not kept up with modern developments in science, particularly in physics. As a result scientists have become the bearers of the torch of discovery in our quest for knowledge.

As William Lane Craig has remarked, such a verdict is

not merely condescending, but also…outrageously naïve. The man who claims to have no need of philosophy is the one most apt to be fooled by it.

Indeed, Hawking and his sidekick Mlodinow proceed to show just how ignorant they are of philosophy, theology, the philosophy of science, the history of philosophy, the history of science, and general science itself. In the Larry King Live show Hawking was asked who his hero was, and why, to which he responded:

Galileo, the first modern scientist who realized the importance of observation.

Well, you can have who you like as your hero, of course, but the historical claim about Galileo is utter rot. He couldn’t hold a candle to the likes of Kepler, for one. Galileo was a second-rate scientist in the main, who continued to his dying day to deny gravitational force as constraining bodies to rotate around the sun, clinging to an Aristotelian idea that celestial bodies ‘naturally’ moved in ‘perfect’ circles because they were not acted upon by a centripetal force, and he refused to accept Kepler’s careful observations and tabulated data that planets were subject to gravitational pull and moved in ellipses. He likewise refused to believe that the sun and moon caused the tides, as Kepler showed, because he denied extraterrestrial gravity. Apart from his last work, under house arrest, on mechanics, the myth of Galileo’s supposed greatness is the deliberate invention of atheists, communists and other anti-Christians, who have cunningly warped history since the nineteenth century to promote a ‘conflict thesis’. Mighty interesting that Hawking, who has built his reputation on pushing cosmic gravity into the absurd, without observational corroboration, should have as his hero one who denied extraterrestrial gravity and who often espoused dogma over meticulous observation.

But if philosophy is dead, it is dead only in the mind of Stephen Hawking, where it was delivered stillborn, or smothered at birth. As someone has said, when all you have is a hammer, everything looks like a nail. And if ‘scientific’ conjecture is all Hawking has by way of explanation, it does the crudest of jobs, riding roughshod over and mangling all understanding, rationality and logic, so that he ends up making puerile statements unworthy of an intelligent man. Just as, by definition, ‘Intelligent Design’ is not a scientific hypothesis because it deals with causes outside the realm on natural science, likewise a physical explanation cannot be an explanation for a metaphysical problem.

Continue reading ‘Hawking’s Grand Delusion (Part III)’

Hawking’s Grand Delusion (Part II)

Hows, Whys, Wherefores, and Miserable Refuges

Woman teaching geometry, from Adelard's translation of Euclid

[See Part I for introduction]
Adelard of Bath (or Athelhard, AD 1080-1160) is sometimes known as the first English scientist. In his classic work Natural Questions he states:

I will detract nothing from God; for whatever is, is from him and by him; yet not even this is said vaguely and without due care, as we must listen to the very limits of human knowledge: only where this utterly breaks down, should we refer things to God.

In common with Christians down through the ages, Adelard sought natural answers to natural questions as far as such studies could be taken. Natural Questions is a dialogue between Adelard and his nephew in which he asks, ‘Why is there a rainbow in the heavens?’ His nephew replies that it is a sign of God’s promise not to flood the entire earth again. Adelard says

Of course that’s what God said and of course God put the rainbow there, but that doesn’t explain the rainbow. That is an example of a miserable refuge from a real philosophic explanation…I know God did it! But that’s not natural philosophy [i.e. science], that’s theology.

Continue reading ‘Hawking’s Grand Delusion (Part II)’

Hawking’s Grand Delusion (Part I)

Stephen Hawking 'speaks' once again 'ex cathedra'

We consider the 2010 book The Grand Design by Stephen Hawking and Leonard Mlodinow (screenwriter for Star Trek: the Next Generation), but first we must lay some groundwork. For ease of digestion this post is split into three parts, the first two parts being introductory.

Intelligent Design and the limits of science

To start with, here’s an old chestnut: is ‘Intelligent Design’ a scientific hypothesis? Well, it is a hypothesis, and a most intelligent hypothesis, held by the brightest of minds for thousands of years, that something with the appearance of design (which even atheists admit) is actually designed. Whether or not it is true, it cannot be denied except by the most churlish that the inference is a reasonable one. However, if we deliberately limit the term ‘scientific’ to natural science, wherein scientific hypotheses have natural explanations exclusively in terms of natural phenomena from within the natural world itself – a closed system where there is no external causation, or where at the very least external causation is beyond the scope of scientific explanation – then according to this definition intelligent design cannot be a scientific hypothesis.

But so what? All this means is that science is deliberately limited in explanation, and deliberately so limited by definition. Primary causation is not only outside but also incomprehensible to scientific enquiry, so primary causation, even if true, cannot offer a ‘scientific’ explanation. Without access to the designer’s original plan, as it were, where could the hypothesis of intelligent design take us from a ‘scientific’ perspective? It has no explanatory power, no predictive capability, no falsifiability within the self-defined and self-limiting ‘scientific’ realm. As an example, if I tell you in all truth that the jet engine was designed by Frank Whittle, what does that fact tell you about the jet engine other than that it was designed by Frank Whittle?

Continue reading ‘Hawking’s Grand Delusion (Part I)’


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