I believe both temporal and spatial dimensions are virtual in the mind, that is, mental time and mental space. — BrianW

Then we simply disagree--I believe that space-time is a real continuum; i.e., it is as it is regardless of how anyone thinks about it.

This can be on the same object thus making it possible to maintain a static frame of reference for the object distinctly. — BrianW

How is that relevantly different from marking two points in time in order to measure duration?

Can time by itself be considered without any other dimension? — BrianW

Yes--at least from a phenomenological standpoint, thinking has temporal extension but no spatial extension. That is one way to differentiate the mental from the physical.

As stated in the other thread, the observed (and surprising) fact C is a conclusion that deductively follows from A. In Peirce's own words, also from 1908:

At length a conjecture arises that furnishes a possible Explanation, by which I mean a syllogism exhibiting the surprising fact as necessarily consequent upon the circumstances of its occurrence together with the truth of the credible conjecture, as premisses. On account of this Explanation, the inquirer is led to regard his conjecture, or hypothesis, with favor. As I phrase it, he provisionally holds it to be "Plausible."

Peirce's favorite example was Kepler's series of hypotheses regarding the orbit of Mars. After only a few unsuccessful conjectures, he tried an ellipse, which not only fit the data that he had from previous observations, but also led to predictions that were subsequently corroborated by further observations.

A well-prepared mind. Again, Peirce called it a conjecture long before Popper did.

Over the chasm that yawns between the ultimate goal of science and such ideas of Man's environment as, coming over him during his primeval wanderings in the forest, while yet his very notion of error was of the vaguest, he managed to communicate to some fellow, we are building a cantilever bridge of induction, held together by scientific struts and ties. Yet every plank of its advance is first laid by Retroduction alone, that is to say, by the spontaneous conjectures of instinctive reason. — Peirce (1908)

Time ... is dependent on at least two distinct values for its delineation. — BrianW

Again, how is each dimension of space any different in that regard? You need to mark at least two points in order to measure linear distance.

For example, length, surface area, volume and force can be considered in a static state or a static frame of reference. Can the same be said of time? — BrianW

A static state is a hypothetical construct in which we examine the three dimensions of space without considering time. We can likewise omit one spatial dimension and evaluate how a hypothetical two-dimensional state changes over time. We can also omit both time and one spatial dimension for static analysis of a hypothetical two-dimensional state; in fact, this is a very common simplification in my field of structural engineering.

There is some measurable quantity we call time that changes. — Devans99

No, time is not an independent "thing" that changes, it is the (fourth) dimension of space-time that corresponds to spatial change. As I keep pointing out, motion through continuous space-time is a more fundamental reality than discrete positions in space or moments in time, which we arbitrarily mark for the sake of measurement and analysis.

I don't see how you can have a 'durationless instant' surely a contradiction in terms? — Devans99

Exactly--time is not composed of durationless instants, and space is not composed of dimensionless points. Those are human constructs, which are very useful for certain purposes, but not real.

Even when the object is at rest there would still be forces influencing it, generating and maintaining shape/form, generating attraction and repulsion, etc. — BrianW

There is no such thing as an object at rest. Continuous motion through space-time is a more fundamental reality than discrete positions in space or moments in time, which we arbitrarily mark for the sake of measurement and analysis. Where there is no acceleration, there is no force.

I don't know about time, I find it to be purely relative. Can a dimension be relative? — BrianW

What exactly do you mean by "relative"? How is each dimension of space different from time in that regard?

Popper argued that it is logically impossible to verify or falsify a scientific theory. He sets out a Method, based on certain epistemological truths he discovered. That's why it's called the Scientific Method. — Inis

Charles Sanders Peirce spelled out the scientific method (as outlined here) the year after Popper was born.

Or "now"--like any other durationless instant--is simply an arbitrary human construct that marks continuous space-time, rather than a real constituent of time itself.

Personally, I consider the fourth dimension to be that of force. — BrianW

Force is defined as the product of mass and acceleration, which is the second derivative of space with respect to time, so it is not an additional dimension. I often startle young structural engineers right out of school when I tell them that force does not actually exist--it is merely a mathematical construct that enables us to analyze and solve problems.

Frankly, I am surprised that no one has already pointed out that time is widely considered to be the fourth dimension, since space-time is a continuum. So the question is really whether it is possible to imagine a fifth dimension.

The scientific method (TSM) is the fallacy of affirming the consequent.
1) If hypothesis A is true then predictions B, C, D, etc. are true
2. Predictions B, C, D, etc. are true
Therefore
3) Hypothesis A is true
Is the above the correct blueprint for all scientific arguments?
If it is then the fallacy of affirming the consequent has been committed. — TheMadFool

Only if one insists that deduction is the only valid form of logic. Charles Sanders Peirce recognized that retroduction (or abduction) and induction are also valid, just not in the same way as deduction. The argument outlined above is a valid retroduction, which is the first step in any scientific inquiry, with some qualifications that Peirce included in his own formulation of it:

• The surprising fact, C, is observed.
• But if A were true, C would be a matter of course.
• Therefore, we have reason to suspect that A is true.

A is the hypothesis that would explain C. Unlike deduction, the truth of the conclusion is not guaranteed by the truth of the premises; the explanatory hypothesis is merely plausible, rather than certain. The second step in any scientific inquiry is then deduction, deriving additional necessary consequences if the hypothesis is indeed true--i.e., predictions. The third step is then induction, conducting experiments and/or performing additional observations to ascertain whether those predictions are corroborated or falsified. The validity of induction is experiential rather than formal, grounded in the nature of its method--it is self-correcting in the long run.

...and the difference between "real" and "actual" is...? — Pattern-chaser

The real is that which is as it is regardless of what any individual mind or finite group of minds thinks about it. The actual (or existent) is that which reacts with other like things in the environment. Hence reality and actuality are not coextensive--besides real actualities (e.g., individual events), there are also real possibilities (e.g., qualities) and real conditional necessities (e.g., laws of nature) that cannot be reduced to collections of their actual instantiations.

A few quotes from Charles Sanders Peirce seem relevant here.

• "Just as we say that a body is in motion, and not that motion is in a body, we ought to say that we are in thought, and not that thoughts are in us."
• "Thinking always proceeds in the form of a dialogue--a dialogue between different phases of the ego."
• "It is not merely a fact of human Psychology, but a necessity of Logic, that every logical evolution of thought should be dialogic."
• "All thinking is dialogic in form. Your self of one instant appeals to your deeper self for his assent."
• "Even in solitary meditation every judgment is an effort to press home, upon the self of the immediate future and of the general future, some truth. It is a genuine assertion, just as the vernacular phrase represents it; and solitary dialectic is still of the nature of dialogue."

Zeno's paradox even applies to each of the infinitesimally small elements of a supposed hyper-task. — sime

The paradox arises from treating space and time as composed of discrete elements of any kind, rather than recognizing space-time as a true continuum, such that motion is the fundamental reality--not positions or instants, which we arbitrarily mark for the purpose of measurement and analysis.

You said charitable reading is your default...but its not, you just showed it isnt. — DingoJones

I stated quite plainly that I was talking about apparent contradictions across a much larger text, especially one that has been carefully scrutinized by scholars for centuries. If you think that you can judge and dismiss me on the basis of one brief exchange on an Internet forum ... well, cheers.

As I said before ...

If I came to believe that you were sincerely seeking the truth, and thus open to changing your mind, I would be glad to discuss others ... Anyone can quote and compare words written in just about any literary work of significant length to create alleged contradictions. Studying the text as a whole and trying to reconcile them is hard work. — aletheist

It is not a case of contradiction but there is no charitable interpretation to give it. — Andrew4Handel

I have explained (several times now) what I mean by the principle of charity in this context--treating consistency as the default interpretation and attributing actual contradictions only as a last resort.

You have yet to refute one contradiction. — Andrew4Handel

I addressed two alleged contradictions in my very first post in this thread. If I came to believe that you were sincerely seeking the truth, and thus open to changing your mind, I would be glad to discuss others.

The reason I believed it is because it quoted and compared the words written in the bible. — Andrew4Handel

Anyone can quote and compare words written in just about any literary work of significant length to create alleged contradictions. Studying the text as a whole and trying to reconcile them is hard work.

Why wouldnt you assume that my sentence would not be contradictory once you understood its broader context? — DingoJones

In this particular case, because you were clearly trying to generate an obvious counterexample in an effort to disparage my purported approach. If I came across that same sentence within a poem, I would be inclined to evaluate it differently. Would you like to suggest some other circumstances in which it would make sense?

So if I write “i like blue skies rather than grey skies, but I like grey skies much more than blue skies”, you would default to the assumption that I easnt being contradicting? — DingoJones

Perhaps you have multiple personalities with different subjective preferences. More seriously, that is an obvious contradiction within the same sentence; I was talking about apparent contradictions across a much larger text, especially one that has been carefully scrutinized by scholars for centuries.

I am not stating my own beliefs I have quoted the scripture. — Andrew4Handel

You are repeatedly stating your dogmatic belief that the Bible is contradictory. Even if I were to offer plausible resolutions for all 492 alleged contradictions on that website in which you evidently have placed your faith, it seems unlikely that you would change your mind.

How do you apply the principle of charity to this verse? — Andrew4Handel

What is the alleged contradiction?

This a great succinct illustration of biblical contradiction. — Andrew4Handel

Only if one is easily persuaded by shallow caricatures.

I am referring to what the bible actually says and not Christians believe. — Andrew4Handel

On the contrary, you are repeatedly asserting your own dogmatic beliefs about the Bible. Why start the thread at all, if your mind was already made up?

Again, when we encounter an apparent contradiction in any piece of writing, I advocate being a charitable reader, treating consistency as the default interpretation and attributing actual contradictions only as a last resort.

It is not alleged contradictions. — Andrew4Handel

I explained why two of the alleged contradictions are not actual contradictions. As you said yourself ...

If someone wants to refute the claim that the bible contradicts itself then they can prove that by refuting a given example of a contradiction. — Andrew4Handel

I am obviously not going to take the time to deal with all 490 other alleged contradictions. My point was mainly to illustrate my approach to what you supposedly wanted to discuss in this thread.

But what concerns me here is when you do have a contradiction how you maintain or discover the truth. — Andrew4Handel

When you encounter an apparent contradiction, I advocate being a charitable reader, which means treating consistency as the default interpretation and attributing actual contradictions to any piece of writing only as a last resort.

In any case, as I said before, your mind already seems to be made up--i.e., your current beliefs about the Bible appear to be just as dogmatic as those of any fundamentalist.

Then as a young adult I discovered a website concerning numerous contradictions in the bible. — Andrew4Handel

As Abraham Lincoln famously said, "Do not believe everything that you read on the Internet."

More seriously, as noted, "contradiction" has a very specific definition in logic and philosophy; it requires the assertion of both A and not-A at the same time and in the same sense. I suspect that you would find, upon careful examination, that most of the alleged contradictions in the Bible would not actually qualify as such in this rigorous sense. In fact, theologians over the centuries have recognized them, wrestled with them, and offered plausible resolutions. For example ...

Issues such as are there unforgivable sins
http://skepticsannotatedbible.com/contra/unforgivable.html
and who should the gospel be preached to
http://skepticsannotatedbible.com/contra/samaritans.html — Andrew4Handel

Regarding the first issue, the consensus among Christians is that the only unforgivable sin, which Jesus called "blasphemy against the Holy Ghost," is rejecting the forgiveness that God otherwise offers. Regarding the second issue, the restrictions that Jesus imposed on His disciples (Matthew 10) and that the Holy Ghost imposed on Paul (Acts 16) were clearly intended only for those specific occasions, not applicable for all time.

I am not really interested in trying to deal with every alleged contradiction in the Bible, especially since your mind already seems to be made up. Hopefully this just gives you an idea of how an intentionally charitable reading--one that treats consistency as the default interpretation, rather than contradiction--can be helpful for understanding why many very intelligent people can and do hold it in such high regard.

/thread — S

If only.

'when you change something, it is changed'.
Infinity in mathematics does not follow this axiom. — Devans99

and already explain how this is false. The axiom that you are really following is, "When you add a finite quantity to another finite quantity, you get a different finite quantity." That axiom straightforwardly does not apply to infinity, which does not entail that infinity is somehow contradictory--only that it is different from a finite quantity, and must accordingly be treated differently than a finite quantity.

Thanks--all of that is consistent with my understanding, as well. :up:

The problem is infinity does not follow the axiom: 'if I add (non-zero) to something, it changes'. Thats such as basic axiom, taken as reality by most people... — Devans99

The problem is that relativity does not follow the axiom: "no matter how fast something is traveling, mass, length, and time are constant." That is such a basic axiom, taken as a reality by most people ...

The problem is infinity does not follow the axiom: 'if I add (non-zero) to something, it changes'. Thats such as basic axiom, taken as reality by most people... — Devans99

Because most people only ever deal with and think about finite quantities, which is the domain in which that axiom applies.

Ah, we finally get to the heart of the matter--it is not that the definition of infinity is contradictory, as the thread title asserts, but that you do not like including something in mathematics that does not follow the same rules as finite quantities.

X+1=X never occurs in maths, apart from when it comes to infinity. — Devans99

Indeed, infinity is different from any finite quantity. So what? That does not make it illogical or contradictory, just different.

... the rest of maths obeys the arithmetic operators (or appropriate variations of them), infinity should too. — Devans99

Why? The fact of the matter is that it does not, so we can either throw up our hands (like you do) or find and develop meaningful alternatives (like mathematicians have).

I challenge you to come up with another mathematical 'number' that you can add a non-zero amount to without changing? — Devans99

Why is another example required to justify the one that we have been discussing? The whole point is that the mathematics of finite quantities are (rather obviously) not applicable to infinity.

That is fair. Can you elaborate on how QM supports the continuity of space-time? What is your interpretation of the Planck length and Planck time?

There are more numbers than there are square numbers yet each number has a square. — Devans99

How many numbers are there? How many square numbers are there? Unless you can answer those two questions, you cannot assert that one is greater than the other. Note that we are not talking about any finite interval, we are talking about all numbers and all square numbers.

We know by induction that there are more numbers than square numbers in all finite intervals so we can induce this implies to infinity as a whole. — Devans99

See, the only thing contradictory in this entire discussion is your childish insistence on repeatedly applying the axioms of finite mathematics to infinity. Your "induction" here is straightforwardly false.

BTW A paradox is usually indicative of an underlying logic error. — Devans99

Indeed, an underlying logic error by the person who thinks that a paradox entails a contradiction.

Space and time. I observe and experience them, and our best models of them require the assumption that they are infinitely divisible. — MindForged

Do the assumptions underlying our best mathematical models of something qualify as observations and experiences of the real object itself? Our best mathematical models of buildings for structural analysis consist of finite elements, but no one would seriously claim that we observe and experience real buildings as collections of finite elements.

In any case I would suggest that space-time is an example of observable continuity, rather than observable infinity; and since the concept of infinite divisibility has proven problematic in past discussions, I would suggest infinite magnification as an alternative. No matter how much you were to "zoom in" on space-time, you would always "see" a four-dimensional continuum, all the way down to the infinitesimal level; never a discrete point at a discrete instant.

We don't produce axioms in geometry to measure things, that's just a very useful feature of geometry. — MindForged

Exactly, and the same is true of mathematics in general. We generate formal hypotheses and work out their necessary consequences, only some of which turn out to be useful for measuring or otherwise analyzing actual phenomena. That is precisely why we make a distinction between "pure" and "applied" mathematics.

How can a quantity not change when you add another positive quantity to it? Thats impossible so infinity is not a quantity. — Devans99

Once again, you are smuggling in an additional premise--in this case, that something must be a quantity in order to qualify as a number.

I know plenty about the maths of infinity thank you. I have spent much time reading up on it. It's shot through with contradictions and paradoxes. — Devans99

Unfortunately, reading and knowledge do not necessarily translate to understanding. You have yet to identify a single contradiction when the relevant terms are defined consistently, and a paradox is simply an opportunity to think more carefully.

Math is hard. — Rank Amateur

So is logic, apparently; among other things, it requires being explicit about one's premises and consistent in one's use of terminology and definitions.

An infinite number is a number bigger than any number... same contradiction. — Devans99

An infinite number is a number bigger than any assignable quantity or countable number ... no contradiction.

Logical concepts only I would argue should be in maths. — Devans99

Your original statement implied that only numbers belong in mathematics, so this is an improvement.

1+∞ = ∞ implies 1 = 0 is not logical. — Devans99

What is not logical is the claim that "1+∞ = ∞ implies 1 = 0"; it reveals an utter lack of understanding about the mathematics of infinity, which at this point is clearly willful.

If a number is neither assignable or countable; then what sort of a number is it? — Devans99

The author of the definition that you quoted would presumably reply: It is an infinite number.

It is not a number. — Devans99

Once again, you are smuggling in an additional premise--in this case, that something must be assignable or countable in order to qualify as a number.

Infinity can’t be a number. So it is not maths. — Devans99

A triangle cannot be a number. Does that mean geometry is not mathematics?

So that is:
- It's a number
AND
- It's greater than any number
The two are contradictory. — Devans99

Read the definition that you quoted more carefully. It does not state, "A number greater than any number," which would indeed be contradictory. Instead, it states, "A number greater than any assignable quantity or countable number," which is not contradictory at all.

Infinity can’t be a number. So it is not maths. — Devans99

A triangle cannot be a number. Does that mean geometry is not mathematics?

I am afraid that I cannot make heads or tails of your first paragraph because of the confusion it exhibits regarding the meaning of terms--infinite, qualitative, quantitative, mathematics/mathematical, category.

You insistently claim that numbers have no actual existence. — Metaphysician Undercover

Yes, but I claim just as insistently that numbers are nevertheless real, because ...

I guess you use "real" in another way, to allow for something which is real, but cannot interact with our world. — Metaphysician Undercover

I have stated this explicitly and repeatedly--I deny that reality and actuality/existence are synonymous. Reality consists of that which is as it is regardless of what any individual mind or finite group of minds thinks about it. Actuality/existence is that which reacts with other like things in the environment. Reality includes some possibilities and some conditional necessities that may or may not ever be actualized.

And "actual possibility" implies that the possibility is interacting with the world, but this contradicts what you've already claimed. — Metaphysician Undercover

There is no contradiction. Something is logically possible if it is merely capable of representation; something is actually possible only if it is also capable of actualization.

You can find this mathematics right in your mind. It's really there, and actual. An hypothesis has actual existence whether or not you believe it to be true. — Metaphysician Undercover

As usual, this reflects conflation of the real with the actual.

Numbers are conceptual and infinite. — Metaphysician Undercover

If numbers are infinite, and mathematics is actual, then I guess there is such a thing as an actual infinity after all. Right?

Now we have the platform for Zeno-type paradoxes between the mathematical concepts of space and time, and the observational concepts. What do you think is the appropriate procedure to resolve the incompatibility? — Metaphysician Undercover

Recognize that continuous motion through space-time is a more fundamental reality than discrete positions in space and/or discrete instants in time. We arbitrarily impose the latter for the sake of measurement and calculation.

To place infinity into the category of ideal, would only remove it from reality, if you proceed like aletheist above, on the preconceived notion that ideas are not real. — Metaphysician Undercover

Where on earth have I ever suggested that ideas are not real? Perhaps this reflects yet another conflation, this time between two definitions of "idea"--the content of an actual thought vs. anything whose mode of being is its mere possibility of representation. The latter is real even if it never actually gets represented, which means ...

I agree with what you have described here, a possibility is defined by actuality, what actually is. — Metaphysician Undercover

... this is incorrect. Possibility is a distinct mode of being from actuality--and from (conditional) necessity, as well; none of them is dependent on either of the others. That is precisely why we must carefully distinguish logical possibility from actual possibility. Mathematics deals with that which is logically possible, regardless of whether it is actually possible.

... the mathematics must somehow interact with things in order that these things get built. — Metaphysician Undercover

Again, where can I find such mathematics so that I may interact with it? We can only interact with that which is actual, which is why both words have the same root; but mathematics deals entirely with the hypothetical. We use mathematics to model the actual, but that is not interacting with mathematics as if it were something that exists.

How is it that mathematics is necessary for building things, yet numbers do not interact with anything? — Metaphysician Undercover

Wow, do you really think that mathematics is necessary for building things? That would be news to the ancients, or to any young child even today who builds things while playing. Mathematics is certainly useful for analyzing, designing, and building things--especially large, complex things--but it is by no means necessary.

I suppose engineering could be done without numbers? — Metaphysician Undercover

I suppose it depends on how you define "engineering." At this stage of my own career as a structural engineer, I spend most of my time making high-level decisions that involve the exercise of practical judgment obtained through experience, rather than crunching numbers.

Obviously numbers interact with things ... — Metaphysician Undercover

Really? Where can I find a number so that I may interact with it?

Of course they're not measurable shades if they're not actual shades, only potential shades. — Metaphysician Undercover

I have consistently characterized a continuum and an infinitesimal as real--that which is as it is, regardless of what any individual mind or finite group of minds thinks about it--but not actual.

So all you are doing is attempting to limit, arbitrarily, our capacity to measure a length, by saying that this length, the infinitesimal length, is such a limit. — Metaphysician Undercover

This is exactly backwards--what is arbitrary is the insistence that anything must be measurable in order to be real.

You mean, like saying that there is a number which has no definite value, but it is nevertheless a number? — Metaphysician Undercover

An infinitesimal is not a number.

That's what gives mathematical objects their actual existence, the definition. To say that there is a mathematical object which is indefinite is nonsense. — Metaphysician Undercover

What is nonsense is claiming that mathematical objects have actual existence at all. In themselves, numbers (for example) are aspatial and atemporal, and do not react to or interact with anything else.

Each of those shades of colour is measurable though. — Metaphysician Undercover

False. Again, between any two measurable shades, there are intermediate potential shades beyond all multitude that cannot be measured, even in principle. That is what it means to be a true continuum.

Since dimensionality constitutes being measurable ... — Metaphysician Undercover

It straightforwardly begs the question to define dimensionality as "being measurable," when what is at issue is the logical (not actual) possibility of dimensionality that is not measurable. Measurement entails discreteness, but we are talking about true continuity.

The point being that you defined infinitesimals as having no specific, or definite, or measurable dimensionality, so it is contradictory to talk about a "one-dimensional infinitesimal". — Metaphysician Undercover

By definition, a one-dimensional infinitesimal has dimensionality, even though it cannot be measured along that one dimension. Its "length" relative to any finite/discrete unit is less than any assignable value, but nevertheless not zero.