As we continue to explore the significance of logic, the pivotal importance of Mathematics (and of the mathemat-ICAL ordering of reality) has come up. Where, we can best understand mathematics in two frames by using a definition with a bracket: Mathematics is [the study of] the logic of structure and quantity.
The study part is cultural, the logic part speaks to an intelligible rational framework inextricably embedded in the existence of a world with distinct identity and then with structures amenable to quantification. So, let us headline a comment from the thread on no 4:
87: >>Let us take a key observation:
There is order in the universe and we are good at modelling it mathematically. But that doesn’t mean that mathematics exists without humans.
Here we see the concept that mathematics is essentially a human practice, a study. The contrast, there is order in the cosmos then uses a shift in terminology that misses a key aspect of that order — it is in material part inherently quantitative and structural. That is, it is of the substance that we may appropriately call mathematical.
Not that the labelling creates the reality as an inner phenomenon locked away from the world of things in themselves by the ugly gulch, that collective labelling (see the slippery slope to solipsism?) is a response to discovery.
We must face that, starting with that once any distinct world is, that means that there is inherently a contrast between some A and what is not A, ~A. Existence as a distinct physical entity or even as an idea immediately has as corollary, distinction from what is not the same. We therefore see existence vs non-existence, nullity. We see existence, unity. We see contrast thus duality. The numbers 0, 1, 2 swim into view, not the labels [numerals] we happen to find convenient, the substance.
[Let’s add, how this leads to the construction of the counting numbers:
{} –> 0
{0} –> 1
{0,1} –> 2
{0,1,2} –> 3
. . .
Where, we may go beyond, to the panoply of numbers great and small, the surreals:]
This surfaces the question, what is truth. Again, not the label but the substance. To which Aristotle’s response speaks: truth says of what is that it is; and of what is not, that it is not. Truth accurately describes reality. It is attainable by us in key cases, e.g. error exists is undeniably true. And as that is so, knowledge exists as warranted, reliable truth that we acknowledge [to know, one has to accept, to actually believe], in this case to utter certainty.
Of course, nowadays, it is a struggle to form a coherent concept of truth, warrant, knowledge. That is how impoverished our sadly suicidally bent civilisation has become. And of course, yes these are going to seem strange and will be hard or even bitter to swallow. That is going to take time.
Now, back to the mathematical substance of reality. We see from distinct identity of a world, that the natural numbers are necessarily substantially present given the force of that identity. But, isn’t all of this just in our heads, if we aren’t here to do math it vanishes for want of sufficiently complex brains, poof.
Nope.
|| + ||| –> |||||
obtains whether we are there to perceive and contemplate it or not. Two is even and the first prime. Three is prime simply as it cannot be evenly shared in a whole number of slots apart from by ones. And so forth, property after property. We come along, discover significance, label, embed in systems of thought. But all of that is in response to substance that we find it necessary to accurately describe.
We develop symbols:
2 + 3 = 5
Those are cultural, convenient, helpful. But they are not the substance, they are our way to handle that substance. A substance of reality that is quantitative (amenable to measure) and structural (bound up in coherent relationships). We label such, Mathematics. That does not create a discipline by the poof-magic of words, it is a way for us to refer to the substantial reality and to our explorations and development.
But that’s just your view.
Nope, views must be accountable to substantial reality. That substance starts with there being a distinct world. We are seeing something that is objectively true, credibly accurately describing realities that are beyond being figments of imagination such as Mr Spock or Fr Brown or Tolkein’s world and rings.
Beyond numbers we may see the continuum, the rock falls through space. But what is space. We see here a quantitative structure tied to length and dimensionality. There is more or less of length and it can be in different directions. To get there conceptually, we need integers (thus negative numbers), we need fractions, we need power series that can sum to infinite numbers of terms and which converge in some cases, as we discovered — nope, it was not an invention — the irrationals. Sqrt-2 deserves to be a famous number once it was seen that the diagonal of a square was inherently in-commensurable with its sides. Imagine, this was actually seen as of such significance that it was viewed with religious awe.
Yes, they saw more truly than we now do in too many cases, this sees into the roots of reality.
Likewise, 0 = 1 + e^i*pi should not only give high confidence in the coherence of vast domains in maths, but it should give us a view into the roots of reality. Another day.
Just to get to understand the space in which a stone falls, we have seen a huge body of quantitative structures embedded inextricably in reality. The substance is there. Bring on the time for a trajectory and we see the same continuum again.
Then, to get to the substance of the gravitation that is at root of the force that triggers the accelerated motion, we find not only further structures and quantities such as displacement x, velocity dx/dt and acceleration dv/dt as well as the panoply of infinitesimals, limits etc, but also the warping of spacetime through the influence of a massive body, a planet. (A ship or a mountain shows similar effects, on a smaller scale.)
Then, looking through telescopes, we find that gravitation is central to the grand structure and function of the overall world we inhabit.
Reality is pervaded with a rationally accessible framework that is quantitative and structural.
Nor does the struggle we may have to acknowledge that pervasive substance change it. It simply forces us to face the need to pursue comparative difficulties and recognise what is a superior explanation. It is simply not good enough to say, oh yes order without acknowledging the specific type: structural, quantitative — that is, mathematical.
And so, we come to Newton, in his General Scholium to Principia; who was moved to an exercise in philosophical theology as he pondered his discoveries:
. . . This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One; especially since the light of the fixed stars is of the same nature with the light of the sun, and from every system light passes into all the other systems: and lest the systems of the fixed stars should, by their gravity, fall on each other mutually, he hath placed those systems at immense distances one from another.
This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God pantokrator , or Universal Ruler; for God is a relative word, and has a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God usually signifies Lord; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God: a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence from infinity to infinity; he governs all things, and knows all things that are or can be done. He is not eternity or infinity, but eternal and infinite; he is not duration or space, but he endures and is present. He endures for ever, and is every where present; and by existing always and every where, he constitutes duration and space. Since every particle of space is always, and every indivisible moment of duration is every where, certainly the Maker and Lord of all things cannot be never and no where. Every soul that has perception is, though in different times and in different organs of sense and motion, still the same indivisible person. There are given successive parts in duration, co-existent puts in space, but neither the one nor the other in the person of a man, or his thinking principle; and much less can they be found in the thinking substance of God. Every man, so far as he is a thing that has perception, is one and the same man during his whole life, in all and each of his organs of sense. God is the same God, always and every where. He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance. In him are all things contained and moved [i.e. cites Ac 17, where Paul evidently cites Cleanthes]; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the omnipresence of God. It is allowed by all that the Supreme God exists necessarily; and by the same necessity he exists always, and every where. [i.e accepts the cosmological argument to God.] Whence also he is all similar, all eye, all ear, all brain, all arm, all power to perceive, to understand, and to act; but in a manner not at all human, in a manner not at all corporeal, in a manner utterly unknown to us. As a blind man has no idea of colours, so have we no idea of the manner by which the all-wise God perceives and understands all things. He is utterly void of all body and bodily figure, and can therefore neither be seen, nor heard, or touched; nor ought he to be worshipped under the representation of any corporeal thing. [Cites Exod 20.] We have ideas of his attributes, but what the real substance of any thing is we know not. In bodies, we see only their figures and colours, we hear only the sounds, we touch only their outward surfaces, we smell only the smells, and taste the savours; but their inward substances are not to be known either by our senses, or by any reflex act of our minds: much less, then, have we any idea of the substance of God. We know him only by his most wise and excellent contrivances of things, and final cause [i.e from his designs]: we admire him for his perfections; but we reverence and adore him on account of his dominion: for we adore him as his servants; and a god without dominion, providence, and final causes, is nothing else but Fate and Nature. Blind metaphysical necessity, which is certainly the same always and every where, could produce no variety of things. [i.e necessity does not produce contingency] All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being necessarily existing. [That is, implicitly rejects chance, Plato’s third alternative and explicitly infers to the Designer of the Cosmos.] But, by way of allegory, God is said to see, to speak, to laugh, to love, to hate, to desire, to give, to receive, to rejoice, to be angry, to fight, to frame, to work, to build; for all our notions of God are taken from. the ways of mankind by a certain similitude, which, though not perfect, has some likeness, however. And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to Natural Philosophy.
Yes, the mathematical substance that we find inextricably intertwined in the fabric of reality may be disturbing and may challenge us to ponder its worldviews significance. Perhaps, that was intended by the One who framed it.
But, that is an onward question, the matter before us is the need to acknowledge that rationally accessible framework of structure and quantity.>>
Let us ponder. END