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Monroe College ​most of us at one point in our childhood had that moment in a math class where we lay down our pencil and said what on earth am i doing am I actually going to use this my everyday life hopefully you're older and wiser you realize that you actually use math quite a bit but mathematicians and philosophers ask different questions about mathematics other than do we really use it they ask questions such as where did that come from was it created or discovered by man is there an underlying objective truth to mathematics or is the only truth we can find it the logic itself my goal is to answer a few of these questions today through three arguments language logic and art language by an argument of analogy shows that mathematics is a model for an underlying metaphysical reality logic drives mathematics towards an underlying truth that initially can seem absurd impossible but as physical manifestations in our world art is linked to mathematics through creativity creativity which in mathematics is aimed what expressing that underlying metaphysical objective truth so the end of my discussion today I will hope that you come to the conclusion that pure mathematics exists objectively metaphysically and abstractly apart from mankind's discovery manipulation and expression of it over 2,000 years ago a Greek man named Euclid release his system of geometry to the world it was revolutionary because it combined a common intuition with logical consulate it was quickly elevated to a position that the highest object hitting and true mathematics from the greco-roman era ons medieval and even into the renaissance and enlightenment now one of the reasons this geometry was so revolutionary was that it was based on only five building blocks five starting points from which you could build everything else these are called postulates the first four postulates were intuitive and easy to understand then we're a straight line can draw between any two points any straight line segments can be extended indefinitely in a straight line and given any straight lines second a circle can be drawn having that lime segments as its radius and all right angles are equal simple intuitive easy to understand however the fifth postulates stuck out like a sore thumb compared to the previous four for it was much more convoluted and much much less intuitive it went as follows if two lines are drawn which intersect a third in such a way that some of the inner angles on one side is less than two right angles and the two lines and that we must respect each other on that side of the extinct far enough worried convoluted it's much different than the previous four and this gave mathematicians great pause because of his entire system of geometry which was based on intuition had one of the foundation points as something that wasn't intuitive at all so for over a thousand years after tations tried to prove them to the posh hood based on the first four because they could use in two ish they try to use that logic now they couldn't do it it was impossible finally in 1800s two mathematicians on separate side of the European continent did what no one had thought to do before they cast aside the fifth postulate and accepted only the first four is truth this created new non-euclidean geometries but what shocked the mathematical world was that these geometries actually had physical manifestations with a reality so the type of mathematics the geometry that had been held up as the most objective the most true in mathematics for over a thousand years was shown to be only one of the correct geometries given any certain circumstance this gave rise to our sense of relativism within mathematics what if there an infinite number of geometries recreate that we're all equally true this gave rise to mathematical philosophy and formalism then mathematics only meaning can be found in the logic itself it is no outside objective meaning this is in contrast to the mathematical plainness viewpoint which is the view that mathematics exists metaphysically outside our physical world and is based on absolute truth and mankind discovers it rather than creates it Michael today is to reconcile our experience with mathematics using a distinction between pure and applied mathematics pure mathematics being the abstraction of the symbolism of mathematical functions from our physical world and applied mathematics being the actual application of that mathematics physics engineering etc through the three arguments that is earlier language logic language is a model for reality let me explain if I'm having a conversation with an individual in a foreign language I've been doing so in one of two ways I can be hearing the words they're foreign tongue and immediately associating it with a concept or idea that's if I'm fluent in the language but if I'm not as flew into the language that I'm hearing the words the foreign stuff translating those words into English words within my head and then associating it with a concept or idea there's no meaning of the words themselves until we assign them and underlying reality let me explain further so is chair up here if I were to say the word chair a concept of chair would pop into your mind there's nothing special about the sounds check and here it's when they're all mashed together in a certain order and a certain amount of time they bring a concept of chair to mine if I gave you the words Percy sessle SIA toggle angulo those would be nothing to you but if you were from Indonesia Spain Kenya these words would bring forth the concept of cherry thus

our words and sounds have no objective being until we assign them to an underlying reality same is true the mathematics take the concept of two I can say to you I can hold up two fingers I can write a Roman numeral to an Arabic numeral to all of these methods would convey the concept of two to you there's no actual significance symbol itself until we assign of the underlying reality of the concept of to the same is true for mathematical equations there's nothing special about the equation one plus one equals two it's not until we recognize the underlying truth that one object and another object combined makes two objects now if the mathematics is a method of expression of symbolism for an underlying reality what is that underlying reality let me explain it this way most of you have seen my karate mobile out of parking lot at one point or nine so that karate mobile exists materially objectively independent from anything to perpetuate its existence now I know a bunch of you in this room are really jealous of that crime what are you guys over here with that so your jealousy of my karate mobile is abstract it's immaterial it's dependent upon your lines to perpetuate its existence it's subjective so mathematics exists like my karate mobile and that is objective and exists independently from any of those to perpetuate its existence but it exists like your jealousy of my car automobile in that it is abstract and immaterial it exists like neither of these because it is eternal and unchanging the karate will be will one day break down and you're jealous you'll go away once you see connects spritely Ralph and run so that language has shown us that mathematics is a model for an underlying metaphysical reality now let us exactly logic logic drives mathematics towards ins that we initially think are absurd when possible but once we stop to look around we realize that led us to a position of truth that as physical manifestations in our world take imaginary numbers for over a thousand years mathematicians are problems with their equations they ran into situations where they had to take the square root of a negative number now this breaks the rules of algebra can't be done so the mathematician has just had to stop at this point they couldn't do anything else until the 16th century an Italian mathematician named rafael bombelli chose to formalize some rules for dealing with these numbers and that 50 years later Rene Descartes chose to further solidify these rules and dubbed them imaginary numbers he meant this in a derogatory way that there's no way these numbers could possibly exist even break lines after a day de cartes de thought these numbers couldn't exist such as Isaac Newton who believed imaginary numbers were absurd however today after the work of gossip Euler we realize that they can be found everywhere in our world we couldn't have this building we're in today nor the electric libel or the air conditioning or the sound system you're hearing me through without an understanding of imaginary numbers what governs our understanding everything from fluid dynamics to electricity to computing to even the physical force exerted on steel beams this is just an example of one time in logic drove us towards a point that we thought was impossible the probable but once we stopped to look around we realized there was truth in it that had physical manifestations in our world goblin fraga appointed put it this way to discover truth is the task of all sciences it falls to logic to discern the laws of truth this is in direct conflict with the mathematical formulas viewpoint that the only truth that can be assigned to mathematics is the logic itself there's no objective truth outside but as we've seen logic drives and mathematics towards a position of truth that although initially can sometimes think it's absurd is there as manifestations in that real world now let us examine art art is the embodiment of creativity within our culture if I secret creativity you will immediately think of art now our creativity seats into every aspect of our lives from cooking to business to driving to you guessed it mathematics if you ever saw an equation for you how to use a certain amount of creativity and ingenuity to do so knowing which rules which equations to use win and where and this creativity of mathematics as eggs weren't expressing that underlying metaphysical reality let me explain this way imagine item painting of a tree up here with me that I just painted it's the painting of a tree that I saw we could go in the park now this tree that I painted is a man rate representation of the actual tree I saw a week ago now a certain amount of creativity went into the process of creating this painting but I didn't create the actual tree I just created an expression of the tree mathematics is the same way when we solve equations when we express mathematics we're using creativity but we're creating an expression of the underlying reality we're not creating mathematics itself however many people can mistakenly look at the creativity involved in mathematics and think it is the creation of mathematics itself there's being done gh Hardy said a mathematician like a painter or a moment as a maker of patterns if his patterns are more permanent than theirs is because they are made with ideas plus mathematics is not just about logical continuity it's about problem-solving ingenuity and creativity and this creativity as i said is often mistaken for the actual creation of mathematics now two philosophers that held this viewpoint that mathematics is actually created our chronic ER and dedicated Kronecker said that God made the integers all else is the work of man this yes he acknowledged that it existence of God but he chose to believe that man created the rest of mathematics thus there would be no actual meaning behind the rest of the end of mathematics because we created them it wouldn't be present in our universe dedica took this even further to say numbers are the free creation of the human mind thus he believed that the very building block of published mathematics is built is just

subjective to man's wins we recreated the mathematics itself there's no objective meeting aside from the logic itself and this is highly indicative of that formulas beautiful I discussed earlier thus art is linked to mathematics through creativity creativity which his aims were expressing that underlying metaphysical objective truth but which oftentimes is mistaken for the actual creation of mathematics she can be seen in the viewpoint of mathematical formalism I want to address the mathematical formulas about mathematical formalism is in the primary philosophical point of mathematical Platonism for the last 150 years one of the greatest works of mathematical formalism was the pooka via Mathematica published in the early part of the 20th century it was written by Bertrand Russell and Alfred Whitehead who continued David Hilbert one of the fathers of mathematical formalism his work to create a self-contained logical system that was consistent and all-encompassing now just after the rickety of mathematic was published Kurt Goodell a German mathematician published his logical proofs called his incompleteness theorems that pointed out major holes in this formulas viewpoint that they could actually do what they said they can do he pointed out that no arithmetic system could be consistent if asserted its own consistency he also proved that no arithmetic system could be mathematically all-encompassing you could have a comprehensive mathematic system there'd be always something you could neither prove nor disprove and this not two major holes of what the formless believe they can do with this logical system for what they believe they can do is create a self-contained consistent logical system that was recom prehensile this knock out to of the major points with a miss you Boyd now aside from the logical aspect of the weakness of mathematical formalism let's look at the experience we had within mathematics as a viewpoint what physicists and engineers perform mathematical equations they're not approaching the situation in that they're starting with certain starting rules and just seeing where their logic leads and from there they might be successful they might not be for one of the viewpoints of formalism is that mathematics is the self-contained axiomatic system where because mathematics is no objective meaning then the equations that come up with can't be true they're merely successful equations thus mathematics are just a game they play and this isn't the way physicists nor engineers approach their problems they're looking for a truth which they can apply and repeat over and over again and use discover more truths to build on the truth they have discovered that has physical manifestations in the world how can we have gone to the moon or created hundred-foot skyscrapers or even created the computer without the mathematicians behind a thing believing they can solve this problem believe they can find this underlying truth to give them this thing thus mathematical formalism has been shattered or at least severely limited and what it can say it can do thus in conclusion lightly she showed us that argument of analogy that mathematics is a model for an underlying metaphysical and absolute truth logic drives mathematics towards this truth although we can initially think that is absurd or impossible but these truths have that infestations physical world and art is linked to mathematics through creativity creativity which is aimed toward expressing the underlying metaphysical truth of mathematics so in conclusion pure mathematics exists that physically and objectively and abstractly apart from venkata manipulation expression and discovery of it thank you very much St. John Fisher College, Pittsford.

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