Physicist: In this case there’s no proof. With the exception of 0 and 1, all numbers are defined in terms of simpler numbers. “4″ is Defined as “1+1+1+1″. And “2″is Defined as “1+1″.
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Recent
 Q: Before you open the box, isn’t Schrödinger’s cat alive or dead, not alive and dead?
 Q: How many times do you need to roll dice before you know they’re loaded?
 Q: Since it involves limits, is calculus always an approximation?
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 Q: Where do the rules for “significant figures” come from?
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 Q: Is there a formula for finding primes? Do primes follow a pattern?
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 Q: How do you define the derivatives of the Heaviside, Sign, Absolute Value, and Delta functions? How do they relate to one another?
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 Dragon*Con 2013
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 The nuptial effect
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 Basic math with infinity
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 Learning intro number theory
 Q: Is the Alcubierre warp drive really possible? How close are we to actually building one and going faster than light?
 Q: Is darkness a wave the way light is a wave? What is the speed of dark?
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 Q: What is quantum teleportation? Why can’t we use it to communicate faster than light?
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 Q: Will the world end tomorrow?
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 Q: What are fractional dimensions? Can space have a fractional dimension?
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 Q: Is it possible to experience different rates of time? If time were to speed up, slow down, or stop, what would you experience?
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 Q: How accurately do we need to know π? Is there a reason to know it out to billions of digits?
 My bad: If fusion in the Sun suddenly stopped, what would happen?
 Q: If fusion in the Sun suddenly stopped, what would happen?
 Q: Does opening a refrigerator cool down the room?
 Q: What is the probability of an outcome after it’s already happened?
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 Q: Why are the days still longer than nights, until a few days after the fall equinox?
 Q: What is a Fourier transform? What is it used for?
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 Q: Is there a number set that is “above” complex numbers?
 Q: Are the brain and consciousness quantum mechanical in nature?
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 Q: What are virtual particles?
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 Q: If Earth was flat, would there be a horizon? If so, what would it look like? If the Earth was flat and had infinite area, would that change the answer?
 Q: Is there an experiment which could provide conclusive evidence for either the Many Worlds or Copenhagen interpretations of quantum physics?
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 Q: How does instantaneous communication violate causality?
 Q: What is the “False Vacuum” and are we living in it?
 Q: How would the universe be different if π = 3?
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 Q: Do colors exist?
 Q: How can we see the early universe and the Big Bang? Shouldn’t the light have already passed us?
 Q: Are beautiful, elegant or simple equations more likely to be true?
 Q: If quantum mechanics says everything is random, then how can it also be the most accurate theory ever?
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 Q: What would the universe be like with additional temporal dimensions?
 The 2012 Venus transit
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 Q: Can you poke something that’s far away with a stick faster than it would take light to get there?
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 Q: Is the quantum zeno effect a real thing?
 π day!
 Q: Is there an intuitive proof for the chain rule?
 Q: How do you write algorithms to enycrypt things?
 Q: Satellites experience less time because they’re moving fast, but more time because they’re so high. Is there an orbit where the effects cancel out? Is that useful?
 Q: Is it possible to objectively quantify the amount of information a sentence contains?
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 Q: Would it be possible in the distant future to directly convert matter into energy?
 Q: What’s the difference between antimatter and negativematter?
 Q: Why does gravity make some things orbit and some things fall?
 Q: Do you need faith to believe in science?
 Q: What keeps spinning tops upright?
 Q: Do time and distance exist in a completely empty universe?
 Q: Why is it that photographs of wire mesh things, like window screens and grates, have waves in them?
 Q: How does quantum physics affect electron configurations and spectral lines?
 Q: Is it possible for an atomic orbital to exist beyond the s, p, f and d orbitals they taught about in school? Like could there be a (other letter) orbital beyond that?
 Q: Will the world end in 2012?
 Q: How do you find the height of a rocket using trigonometry?
 Q: What are chaos and chaos theory? How can you talk about chaos?
 Q: What is the Riemann Hypothesis? Why is it so important?
 Q: Why does the entropy of universe always increase, and what is the heat death of the universe?
 Q: Could God have existed forever? Is it actually feasibly possibly for some ‘being’ to have just existed, infinitely?
 Q: How can wormholes be used for time travel?
 Q: If gravity suddenly increased would airplanes fall out of the sky, or would it compress the air in such a way that airplanes could keep flying?
 Entanglement omnibus!
 Q: How are imaginary exponents defined?
 Q: Why do nuclear weapons cause EMPs (electromagnetic pulses)?
 Q: How does the expansion of space affect the things that inhabit that space? Are atoms, people, stars, and everything else getting bigger too?
 Q: What would Earth be like if it didn’t turn?
 Q: According to the Many Worlds Interpretation, every event creates new universes. Where does the energy and matter for the new universes come from?
 Q: Can wind chill make things “feel” colder than absolute zero?
 Q: What is “spin” in particle physics? Why is it different from just ordinary rotation?
 Q: What is Bayes’ rule and how do I use it to improve my life?
 Q: Are there universal truths?
 Q: What’s the difference between black holes and worm holes? Could black holes take you to other universes?
 Q: Is there an equation that determines whether a question gets answered on ask a mathematician/physicist?
 Q: If you could hear through space as though it were filled with air, what would you hear?
 Q: What is the three body problem?
 Q: How are fractals made?
 Q: CERN’s faster than light neutrino thing: WTF?
 Q: What’s the point of purely theoretical research?
 Q: Why does lightning flash, but thunder rolls?
 Q: Hyperspace, warp drives, and faster than light travel: why not?
 Burning Man 2011
 Q: If light slows down in different materials, then how can it be a universal speed?
 Q: What is mass?
 Q: How much of physics can be deduced from previous equations/axioms?
 Q: If God were allseeing and allknowing, the doubleslit experiment wouldn’t work, would it? Wouldn’t God’s observation of the location of the photon collapse its probability wave function?
 Q: How do those “executive ball clicker” things work?
 Q: Why is cold fusion so difficult?
 Q: Why does light choose the “path of least time”?
 Q: Does light experience time?
 Q: Would it be possible for humans to terraform mars?
 Q: Can light be used to transfer energy instead of power lines?
 Particle physics, neutrinos, and chirality too!
 Q: What are integral transforms and how do they work?
 Q: How does reflection work?
 Q: What does a measurement in quantum mechanics do?
 Q: If you stood in the beam of a particle accelerator, what would happen?
 Q: What exactly is the vacuum catastrophe and what effects does this have upon our understanding of the universe?
 Q: What is a “measurement” in quantum mechanics?
 Q: How close is Jupiter to being a star? What would happen to us if it were?
 Q: Can you fix the “1/0 problem” by defining 1/0 as a new number?
 Q: How can we have any idea what a 4D hypercube or any nD object “looks like”? What is the process of developing a picture of a higher dimensional object?
 Q: Is it possible to destroy a black hole?
 Q: Why does the Earth orbit the Sun?
 Q: If you suddenly replaced all the water drops in a rainbow with samesized spheres of polished diamond, what would happen to the rainbow? How do you calculate the size of a rainbow?
 Q: If we meet aliens, will they have the same math and physics that we do?
 Q: Is 0.9999… repeating really equal to 1?
 Q: What would Earth be like to us if it were a cube instead of spherical? Is this even possible?
 Q: How do velocities add? If I’m riding a beam of light and I throw a ball, why doesn’t the ball go faster than light?
 Q: What is the universe expanding into? What’s outside the universe?
 Cheap experiments and demonstrations for kids.
 Q: How do I estimate the probability that God exists?
 Q: How do you calculate 6/2(1+2) or 48/2(9+3)? What’s the deal with this orders of operation business?
 Q: Is there a single equation that proves black holes are real?
 Q: Is the edge of a circle with an infinite radius curved or straight?
 Q: As a consequence of relativity, objects becomes more massive when they’re moving fast. What is it about matter that causes that to happen?
 Q: What is the evidence for the Big Bang?
 Q: Is there a formula to find the Nth term in the Fibonacci sequence?
 Q: Why is the integral/antiderivative the area under a function?
 Thank you!
 Mathematical proof of the existence of God.
 Video: Getting Computers to Learn
 Q: What is going on in a nuclear reactor, and what happens during a meltdown?
 Q: How do I find the love of my life? (a Mathematician’s perspective)
 Q: Are all atoms radioactive?
 Q: How do you talk about the size of infinity? How can one infinity be bigger than another?
 Q: Why does E=MC^{2} ?
 Q: What are the equations of electromagnetism? What all do they describe to us?
 Q: What is the entropy of nothing?
 Q: How can quantum computers break encryption?
 Q: How does quantum computing work?
 Q: What causes buoyancy?
 My bad: If atoms are mostly made up of empty space, why do things feel solid?
 Q: How many mathematicians/physicists does it take to screw in a light bulb?
 Q: Why is it that (if you exclude 2 & 3) the difference between the squares of any two prime numbers is divisible by 12?
 Q: Why does relativistic length contraction (Lorentz contraction) happen?
 Q: Why does Lorentz contraction only act in the direction of motion?
 Q: If atoms are mostly made up of empty space, why do things feel solid?
 Q: Can we build a planet?
 Q: π = 4?
 Q: How does a scientist turn ideas into math?
 Q: Is Santa real?
 Q: Why isn’t the shortest day of the year also the day with the earliest sunset?
 Q: Why does “curved spacetime” cause gravity?: A better answer.
 Q: According to relativity, two moving observers always see the other moving through time slower. Isn’t that a contradiction? Doesn’t one have to be faster?
 Q: What does 0^0 (zero raised to the zeroth power) equal? Why do mathematicians and high school teachers disagree?
 Q: Can you do the double slit experiment with a cat cannon?
 Q: How is the “Weak nuclear force” a force? What does it do?
 Q: Does Gödel’s Incompleteness Theorem imply that it’s impossible to be logical?
 Q: If accelerating charges radiate, and everything is full of charges, then why don’t I radiate every time I move?
 Q: If you zoom in far enough, what do particles look like?
 Q: What would you experience if you were going the speed of light?
 Q: Why is pi not a definite number?
 Q: What came before the big bang?
 Q: How do “Numerology Math Tricks” work? (adding digits and tricks with nines)
 Q: What is a magnetic field?
 Q: What is the probability that two randomly chosen people will have been born on the same day?
 Q: Which is a better approach to quantum mechanics: Copenhagen or Many Worlds?
 Q: Why is our vision blurred underwater?
 Q: In the NEC “faster than light” experiment, did they really make something go faster than light?
 Q: How does a Tesla coil work?
 Q: What are Feynman diagrams, how are they used (theoretically & practically), and are there alternative/competing diagrams to Feynman’s?
 Q: Does the 2nd law of thermodynamics imply that everything must eventually die, regardless of the ultimate fate of the universe?
 Q: What is The Golden Ratio? How is it used in Mathematics?
 Q: Why can’t you have an atom made entirely out of neutrons?
 Q: What is the physical meaning of “symmetries”? Why is there onetoone correspondence between laws of conservation and symmetries? Why is it important that there is such correspondence?
 Q: Why does energy have to be positive (and real)?
 Q: How does the Twin Paradox work?
 Q: How can photons have energy and momentum, but no mass?
 Q: If you were on the inside of the Sun falling in, the matter closer to the surface doesn’t affect your acceleration, but the matter closer to the core does. Why is that?
 Q: How do surge protectors work?
 Relativity and Quantum Mechanics: the elevator pitch
 Q: Why are orbits elliptical? Why is the Sun in one focus, and what’s in the other?
 Q: What would happen if everyone in the world jumped at the same time?
 Q: How can electrons “jump” between places without covering the intervening distance?
 Q: Why do we only see one rainbow at a time?
 Q: Why does putting spin on a ball change how it moves through the air?
 Quantum mech, choices, and time travel too!
 Q: Why is the speed of light finite?
 Q: Why is the speed of light the fastest speed? Why is light so special?
 Q: Will we ever overcome the Heisenberg uncertainty principle?
 Q: If gravity is the reaction matter has on space, in that it warps space, why do physicist’s look for a gravity particle? Wouldn’t gravity be just a biproduct of what matter does to space?
 Q: Is it possible to beat the laws of physics?
 Q: What’s the chance of getting a run of K or more successes (heads) in a row in N Bernoulli trials (coin flips)? Why use approximations when the exact answer is known?
 Q: Aren’t physicists just doing experiments to confirm their theories? Couldn’t they “prove” anything they want?
 Q: What’s up with that “bowling ball creates a dip in a sheet” analogy of spacetime? Isn’t it gravity that makes the dip in the first place?
 Q: If we find a “Theory of Everything” will we be done?
 Q: Is it possible to say if the Earth is moving or sitting still without going into space?
 Q: Will there always be things that will not or cannot be known?
 Q: If you could see through the Earth, how big would Australia look from the other side?
 Q: How is it that Bell’s Theorem proves that there are no “hidden variables” in quantum mechanics? How do we know that God really does play dice with the universe?
 Q: Does an electric field have mass? Does it take energy to move an electric field?
 Q: What would the consequenses for our universe be if the speed of light was only about one hundred miles per hour?
 Q: Do virtual particles violate the laws that energy can be created or destroyed? Have virtual particles ever been observed? In any other instance can energy ever be destroyed or created?
 Video: How do we know that 1+1=2? A journey into the foundations of math.
 Q: Would it be possible to generate power from artificial lightning?
 Q: What is the optimum spectrum to visualize things with? Theoretically, which type of vision would be the best to see things with?
 Q: What causes iron, nickel, and cobalt to be attracted to magnets, but not other metals?
 Q: Is it possible to fill a black hole? If you were to continuously throw galaxies worth of matter into a black hole, would it ever fill up? And what would theoretically happen if all the matter in the universe was thrown into a single black hole?
 Q: Can math and science make you better at gambling?
 Q: Spectroscopy?
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 Q: If energy is quantized, what is the least amount of energy possible? And how did they measure it?
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 Q: What do complex numbers really mean or represent?
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 Q: How/when will the world end?
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 My bad: Have aliens ever visited Earth?
 Q: How do Bell pairs (entangled particles) behave experimentally?
 Video: What your Spiritual Guru Never Told you about Quantum Mechanics
 Q: How big does an object have to be to gravitationally attract a Human or have a molten core?
 Video: The Scientific Investigation of Aliens – Evidence Examined
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 Q: Who would win in a fight: Gödel or Feynman?
 Q: How hard would it be to make a list of products of primes that could beat public key encryption?
 Q: What are complex numbers used for?
 Q: Can one truly create something from nothing? If matter formed from energy (as in the Big Bang expansion), where did the energy come from?
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 Q: Are there physical limits in the universe other than the speed of light?
 Q: Is it of any coincidence that mathematics is able to describe physical reality – given that both are inventions of the human mind?
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 Q: What’s it like when you travel at the speed of light?
 Q: Is there a real life example where two negatives make a positive?
 Q: Do the “laws” of physics and math exist? If so, where? Are they discovered or invented/created by humans?
 Q: Do we have free will?
 Q: How did mathematicians calculate trig functions and numbers like pi before calculators?
 Q: How can planes fly upsidedown?
 Q: A flurry of blackhole questions!
 Q: Why does going fast or being lower make time slow down?
 Q: What’s so special about the Gaussian distribution (i.e. the normal distribution / bell curve)??
 Q: Is the universe infinitely old?
 Q: Have aliens ever visited Earth?
 Q: Why is the sky blue?
 Q: Is there a formula for how much water will splash, most importantly how high, and in what direction from the toilet bowl when you *ehem* take a dump in it ?
 Q: What is the meaning of the term “random”? Can thinking affect the future?
 Q: Is it possible to choose an item from an infinite set of items such that each one has an equal chance of being selected?
 Q: Do aliens exist?
 Q: Is it true that all matter is simply condensed energy?
 Q: Which is better: Math or Physics?
 Q: Why is the number 1 not considered a prime number?
 Q: If the universe is expanding and all the galaxies are moving away from one another, how is it possible for galaxies to collide?
 Q: What happens when you fall into a blackhole?
 Q: Is the total complexity of the universe growing, shrinking or staying the same?
 Q: If two trains move towards each other at certain velocities, and a fly flies between them at a certain constant speed, how much distance will the fly cover before they crash?
 Q: Why does oxygen necessarily indicate the presence of life?
 Q: What’s the relationship between entropy in the informationtheory sense and the thermodynamics sense?
 Q: Would it be possible to kill ALL of Earth’s life with nuclear bombs?
 Q: Will black holes ever release their energy and will we be able to tell what had gone into them?
 Q: What are the Intersecting Chord and Power of a Point Theorems?
 Q: How far away is the edge of the universe?
 Q: Why do superconductors have to be cold?
 Q: Why does the leading digit 1 appear more often than other digits in all sorts of numbers? What’s the deal with Benford’s Law?
 Q: How does the Monty Hall Problem work?
 Q: How/Why are Quantum Mechanics and Relativity incompatible?
 Q: What the heck are imaginary numbers, how are they useful, and do they really exist?
 Q: What’s that third hole in electrical outlets for?
 Q: Do physicists really believe in true randomness?
 Q: Could a simple cup of coffee be heated by a hand held device designed to not only mix but heat the water through friction, and is that more efficient than heating on a stove and then mixing?
 Q: What did Einstein mean by: “Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”
 Q: Why does saliva boil in the vacuum of space?
 Q: Can things really be in two places at the same time?
 Q: Why do weird things happen so much?
 Q: Why?
 Q: Will CERN create a black hole?
 Q: What’s the highest population growth rate that the Earth can support?
 Q: What is time?
 Q: What is “Dark Matter”?
 Q: Why do heavy objects bend space and what is it they are bending?
 Q: Why does math work so well at modeling the world around us?
 Q: Why is it that when you multiply a positive number with a negative number you get a negative number?
 Q: What is the best way to understand relativity theory? Why is it so counter intuitive?
 Q: Will we ever go faster than light?
 Q: If you were on a space station, would you be able to tell the difference between centrifugal force and normal gravity?
 Q: Is teleportation possible?
 Q: If black holes are “rips” in the fabric of our universe, does it mean they lead to other universes? If so, then did time begin in that universe at the inception of the black hole? Could we be in a black hole?
 Q: Since pi is infinite, do its digits contain all finite sequences of numbers?
 Q: What is the connection between quantum physics and consciousness?
 Q: What is the probability that in a group of 31 people, none of them have birthdays in February or August?
 Q: What is the meaning of life?
 Q: Why is e to the i pi equal to 1?
 Q: How does a refrigerator work?
 Q: How do I find the love of my life?
 Q: Why does “curved spacetime” cause gravity?
 Q: What is monotony?
 Q: How do we know if science is right?
 Q: How plausible is it that the laws of physics may actually function differently in other parts of the universe?
 Q: Are there an infinite number of prime numbers?
 Q: How can we prove that 2+2 always equals 4?
What would happen if the Parallel Postulate became the Parallel Theorem? Would it only effect the scientific community in a hypothetical sense or would there be more drastic implications?
why we not multiply any trigonometry question by zero to solve it..
example prove 2+2 =4
o*(2+2) = 4 * 0
0 = 0
please answer ……….
Although you end up with a true statement (0=0), you don’t get any new information.
For example, if you were stuck with a question like “2x+1=5, what is x?”, you could multiply both sides by zero and get the (true) statement “0=0″, but you still wouldn’t know what x is.
X+X+X+X=(X+X)+(X+X)
4X=2X+2X ………sice; iknow mango=mango=2mango
or 4X=X(2+2)
devide this by X => 4=2+2
or 2+2=4
hence the proof.
sharafali.a, if you’re really intent on providing a proof, you could use the definitions directly instead of involving multiplication, division, distribution, and a variable. 4=1+1+1+1=(1+1)+(1+1)=2+2. Hardly deserves to be called a proof.
vishal, deriving a true fact from an initial statement does not prove the initial statement. This is backwards. I can use that type of argument to “prove” almost anything:
2+2 = 3
o*(2+2) = 3 * 0
0 = 0
In science, theories are often not proven, but disproven. Perhaps the same concept would have to be applied here. If the numbers 0 through 4 are set quantities (so that I can’t just turn around and say, oh let’s make 2=3 (artificially, as in a linguistic change, not a mathematical one)), then by definition a physical total of two 2s would yield four. It may be impossible to prove that 2+2 always yields four, but perhaps it is impossible to disprove 2+2 equalling four ever. Because the physical evidence is overwhelmingly in 2+2=4′s favor, perhaps it is a postulate or observing type of thing, like right angles are always equal, not a proof type of thing. However, applying the Peano axioms, there is a proof. There is one listed here: http://skepticsplay.blogspot.com/2008/12/224proof.html. Whether or not this actually proves 2+2=4 for every case is another matter.
Regardless, unless if you can find an instance where two physical twos never make a physical four (and I don’t mean taking two sets of two black holes and combining them together to see if you end up with four black holes or something to that effect, count before you combine them), 2+2=4.
I like pi(e).
How can we prove that 2+2 always equals 4? Dah, maybe check your report card to see if you graduated from 1ts grade and thus payed attention in school.
How can 2 people each with 2 apples share evenly if each one doesn’t end up with two apples each? Easily, make apple sauce. Thus proving anything can only be possible if we have no constraints.
My 2 cents:
You can prove 2+2=4 if you chose a right “formal system”, that is the right definitions, axioms and the “rules of proof”. And the 2+2 need not be 4… Consider:
a) you are doing modular arithmetics: each time you add two numbers, you take a reminder of the division by 4 as a result. Then,
2 + 2 = 0 (since the reminder of 4 divided by 4 is 0)
Most computers do binary arithmetics like this (just its not modulo 4, but modulo 2^32, or 2^64 nowadays…)
b) you are doing saturation arithmetics — you are only using 4 different numbers, 0, 1, 2, and 3, and whenever a result is larger than 3, it “sticks” to 3:
2 + 2 = 3
This kind of addition is useful in Xray detectors — when a detector is hit by such an intense Xray beam that the intensity exceeds the capacity of the detector, the detector “sticks” to its highest possible reading, and indicates an overflow (which is much better than wrapping back to 0; sure the highest reading is usually not 3 but something like 30 thousand or evem more
3) if you are doing ternary logics where “0″ is “false”, “1″ is “unknown”, “2″ is “true” and “+” is logical “or”, then
2 + 2 = 2
(since if two statements are true, then the statement “either of them or both of them are true” is also true).
d) 2 liters of ethanol + 2 liters of water, when mixed, will not give you 4 liters of solution — it will be slightly less…
e) and finally, when counting apples, sheep, you can reasonably expect that the “usual” school arithmetic
2 + 2 = 4
will apply well…
So basically, what “2 + 2″ amounts to depends on your set of axioms regarding “2″, “+” and “=”. From there on, you can derive a formal, “strict” proof; the reference to Peano arithmetic in a post before was a good demonstration!
Mathematitians are usually interested to derive (or prove) as many theorems as possible from the smallest amount of axioms. But what you prove dependes on what axioms you choose. Physicists are usually interested in mathematics that permits them the best description of the observed world. Which math (i.e. which set of axioms) you choose depends on what kind of phenomena you are invesigating…
Very nice explanation Saulius.
An even harder question: How do you know that 4=1+1+1+1? Why doesn’t 4=1+1? Why was the symbol “4″ assigned to the value of 1+1+1+1?
@Vishal: that is a fallacy of identity. let’s say I change my name to Barack Obama, and it is known that Barack Obama is the name of the President of the United States. Am I then President?
two males in prison and two females in a separate female prison will remain four. a male and a female, two pairs, given the natural order of the world don’t remain only four for very long. life isn’t a series of statics
for the sake of traditional arithmetic proofs, you may need to look at numbers representing sets (church numbers for instance). since the rules of sets follow the rules of arithmetic (mostly) when dealing with countables, arguably you could use sets to prove arithmetic.
according to me it is easy to prove that 2 2=4.If we draw four lines in a plane we can easily see these lines.Now cover two lines by your hand and see two lines.These are two.Now remove your hand and cover last two lines these are also two.Now remove your hand and you can see all four lines.Thus it is proved that 2 2=4.
4=1+1+1+1 and 2=1+1, so 4=(1+1)+(1+1)=2+2 by definition. No need to prove it. 2+2 will not be 4 only if we change the definition for the numbers 2 or 4.
I was playing around with a mathematical system as follows.
0,1 ,+ where a+b=b+a i.e. + is commutative however NOT associative
define successor S(n)=n+1=1+n 2=S(1) 3=S(2) 4=S(3) etc.
so that 2=1+1 3=2+1 4=3+1 (=1+3 also)
furthermore 0 is the “additive identity” so a+0=0+a =a
now 2+2 =(1+1)+(1+1) will be distinct from 4 =(1+1+1)+1 =1+(1+1+1)
Remember + is non associative here.
also 2+3=3+2 will NOT be 5=4+1=1+4
these objects resemble “ordinary numbers but are formally distinct.
This is why I love math. It lets you posit 2+2≠4 counterintuitively.
According to the rules established by us itself, we have got 2+2=4. then how can we go on arguing whether 2+2 actually equals 4 or not????